Clearing a Root Out of a Denominator Makes Your Answer More Accurate
Simplify any surd to its lowest form, add, subtract, multiply and divide surds, kill the root in a denominator with its conjugate, and order surds of different index.
Why bother moving a square root from the bottom of a fraction to the top?
Every textbook insists that a surd must be cleared out of a denominator, and it usually looks like a rule about tidiness.
It is not. It makes your answer more accurate.
Take , and suppose you are working with and , correct to three decimal places.
Divide directly. , and .
Rationalise first. The expression becomes .
The true value is So the direct division is out by about , and the rationalised form is out by about — five times closer, from the same two approximations.
The reason is that dividing by magnifies whatever small error was in the two roots, because a tiny change in a small denominator makes a large change in the quotient. Rationalising replaces a division by a small number with an addition, and addition does not magnify anything.
So the rule exists for a numerical reason and not an aesthetic one — which is worth knowing, because it is the only part of this chapter that is not obvious.
This page covers the second part of the ICSE Class 9 Mathematics chapter on rational and irrational numbers: simplifying surds, the four operations on them, rationalising a denominator with the conjugate, and comparing and ordering surds.
It is not. It makes your answer more accurate.
Take , and suppose you are working with and , correct to three decimal places.
Divide directly. , and .
Rationalise first. The expression becomes .
The true value is So the direct division is out by about , and the rationalised form is out by about — five times closer, from the same two approximations.
The reason is that dividing by magnifies whatever small error was in the two roots, because a tiny change in a small denominator makes a large change in the quotient. Rationalising replaces a division by a small number with an addition, and addition does not magnify anything.
So the rule exists for a numerical reason and not an aesthetic one — which is worth knowing, because it is the only part of this chapter that is not obvious.
This page covers the second part of the ICSE Class 9 Mathematics chapter on rational and irrational numbers: simplifying surds, the four operations on them, rationalising a denominator with the conjugate, and comparing and ordering surds.
How do you reduce a surd to its lowest form?
Pull out the largest perfect square hidden inside the number under the root.
A surd is an irrational root of a rational number — , , . In the number is the order and is the radicand.
A surd is in lowest form when three things are true:
- The radicand has no perfect-square factor other than (or no perfect-cube factor, for a cube root)
- The radicand contains no fraction
- The order is as small as possible
Worked simplifications. Look for the largest square factor each time.
Check it: .
Take out the LARGEST square factor, not just any one. For you could notice and get — which is correct but not in lowest form, because still contains a . Going on gives , the same answer by a longer route.
So if the number still under the root has a square factor, you have not finished. The safe method is to write the radicand as a product of primes: , and every pair of equal primes comes out as one factor — a pair of s gives , the pair of s gives , so comes out and a single stays in.
Clearing a fraction from under the root.
Check it by squaring the answer: .
And a surd's value is unchanged by simplifying, which is the point of the check. and . Simplifying rewrites a number; it never changes it — so squaring your answer, or evaluating both forms, catches every slip in this section.
A surd is an irrational root of a rational number — , , . In the number is the order and is the radicand.
A surd is in lowest form when three things are true:
- The radicand has no perfect-square factor other than (or no perfect-cube factor, for a cube root)
- The radicand contains no fraction
- The order is as small as possible
Worked simplifications. Look for the largest square factor each time.
Check it: .
Take out the LARGEST square factor, not just any one. For you could notice and get — which is correct but not in lowest form, because still contains a . Going on gives , the same answer by a longer route.
So if the number still under the root has a square factor, you have not finished. The safe method is to write the radicand as a product of primes: , and every pair of equal primes comes out as one factor — a pair of s gives , the pair of s gives , so comes out and a single stays in.
Clearing a fraction from under the root.
Check it by squaring the answer: .
And a surd's value is unchanged by simplifying, which is the point of the check. and . Simplifying rewrites a number; it never changes it — so squaring your answer, or evaluating both forms, catches every slip in this section.
How do you add, subtract, multiply and divide surds?
You can multiply and divide any surds of the same order, but you can only add or subtract like surds.
Like surds have the same order and the same radicand — and are like; and are not.
Addition and subtraction. Simplify each surd first, then collect the like ones.
Here and .
Check numerically: , and .
since and .
Check: , and .
Multiplication. Multiply the rational parts and the radicands separately:
Check the second: , and .
Two products worth memorising.
Check the second: , and .
Division.
Check the second: and , and .
Now the restriction that matters most, because it is where marks go. cannot be simplified. It is not .
Check it: , while . They are nowhere near each other.
But multiplication does behave that way: , and .
**So is true and is false. The reason is that a square root distributes over a product and not over a sum** — exactly as squaring does, since while . The two mistakes are the same mistake, seen once with powers and once with roots.
Like surds have the same order and the same radicand — and are like; and are not.
Addition and subtraction. Simplify each surd first, then collect the like ones.
Here and .
Check numerically: , and .
since and .
Check: , and .
Multiplication. Multiply the rational parts and the radicands separately:
Check the second: , and .
Two products worth memorising.
Check the second: , and .
Division.
Check the second: and , and .
Now the restriction that matters most, because it is where marks go. cannot be simplified. It is not .
Check it: , while . They are nowhere near each other.
But multiplication does behave that way: , and .
**So is true and is false. The reason is that a square root distributes over a product and not over a sum** — exactly as squaring does, since while . The two mistakes are the same mistake, seen once with powers and once with roots.
Formula
How does the conjugate clear a surd out of a denominator?
Multiply the top and bottom by the conjugate — the same expression with the middle sign reversed — and the denominator becomes rational by the difference of two squares.
The conjugate of is , and their product is
which contains no surd at all, because the middle terms cancel and the root is squared away.
Worked example 1 — a single surd in the denominator.
Check: , and .
Worked example 2 — a two-term denominator.
Check: , and .
Worked example 3 — two surds.
Check: , and .
Worked example 4 — a surd on top as well.
Check: , and .
Notice that in example 4 the top was multiplied by the conjugate too, which turned it into a square — so the numerator gets longer while the denominator gets shorter. That is the trade, and it is always worth making.
Why it is always worth making, which is the point this page opened with. Once the denominator is rational you can:
- Evaluate the expression from a table of square roots without doing a long division
- Add it to another fraction, because the denominators are now ordinary numbers
- Compare it with another expression
- Get a more accurate numerical value, because you have replaced a division by a small number with an addition
A special case worth spotting instantly. In example 3 the denominator became , and in a case like it becomes — so the whole fraction is simply , with no denominator left at all. **Whenever the two radicands differ by , the conjugate clears the denominator completely**, and that is why the example at the top of this page came out so cleanly.
The conjugate of is , and their product is
which contains no surd at all, because the middle terms cancel and the root is squared away.
Worked example 1 — a single surd in the denominator.
Check: , and .
Worked example 2 — a two-term denominator.
Check: , and .
Worked example 3 — two surds.
Check: , and .
Worked example 4 — a surd on top as well.
Check: , and .
Notice that in example 4 the top was multiplied by the conjugate too, which turned it into a square — so the numerator gets longer while the denominator gets shorter. That is the trade, and it is always worth making.
Why it is always worth making, which is the point this page opened with. Once the denominator is rational you can:
- Evaluate the expression from a table of square roots without doing a long division
- Add it to another fraction, because the denominators are now ordinary numbers
- Compare it with another expression
- Get a more accurate numerical value, because you have replaced a division by a small number with an addition
A special case worth spotting instantly. In example 3 the denominator became , and in a case like it becomes — so the whole fraction is simply , with no denominator left at all. **Whenever the two radicands differ by , the conjugate clears the denominator completely**, and that is why the example at the top of this page came out so cleanly.
How do you decide which of two surds is bigger?
If the orders are the same, compare the radicands; if they differ, make the orders equal first.
Same order — just compare what is inside. For positive numbers, squaring preserves order, so
So because , with no calculation needed.
With rational coefficients, square both. Which is larger, or ?
Since , we get .
Check: and .
Different orders — bring them to a common order. Which is larger, or ?
The orders are and , whose LCM is . Write each as a sixth root:
Now the orders match, so compare the radicands: , hence .
Check: and .
Ordering three of different index. Arrange , and in ascending order. All become sixth roots:
Since , the order is .
Check: .
Evaluating with given approximations. Take , and , correct to three places.
Check the second directly: .
Always rationalise before substituting, and this is the practical rule of the whole chapter. In the first example the direct route gives , while the rationalised route gives — and the true value is .
So the rationalised form is the more accurate one, using exactly the same input data. Subtracting two nearly equal numbers throws away most of the significant figures you had, and then dividing by the small result magnifies what is left of the error.
That is why an examiner asks you to rationalise first and then substitute, and it is the reason the order of the two steps is part of the method rather than a matter of taste.
Same order — just compare what is inside. For positive numbers, squaring preserves order, so
So because , with no calculation needed.
With rational coefficients, square both. Which is larger, or ?
Since , we get .
Check: and .
Different orders — bring them to a common order. Which is larger, or ?
The orders are and , whose LCM is . Write each as a sixth root:
Now the orders match, so compare the radicands: , hence .
Check: and .
Ordering three of different index. Arrange , and in ascending order. All become sixth roots:
Since , the order is .
Check: .
Evaluating with given approximations. Take , and , correct to three places.
Check the second directly: .
Always rationalise before substituting, and this is the practical rule of the whole chapter. In the first example the direct route gives , while the rationalised route gives — and the true value is .
So the rationalised form is the more accurate one, using exactly the same input data. Subtracting two nearly equal numbers throws away most of the significant figures you had, and then dividing by the small result magnifies what is left of the error.
That is why an examiner asks you to rationalise first and then substitute, and it is the reason the order of the two steps is part of the method rather than a matter of taste.
Exam tip
Exam tip: simplify first, then collect — and never add radicands
Take out the LARGEST square factor. If anything under the root still has a square factor, you have not finished. Prime-factorising the radicand is the safe route — every pair of equal primes comes out.
A surd in lowest form has no square factor inside, no fraction inside, and the smallest possible order.
Simplify every surd BEFORE adding or subtracting. looks unlike until you write .
Only LIKE surds can be added — same order and same radicand.
** is NOT .** Check: and . But is true.
**The conjugate of is **, and their product is rational.
Multiply the numerator by the conjugate too — in a case like the numerator becomes a square, so expand it carefully.
Always check a rationalised answer numerically. One substitution catches a sign error instantly.
**To compare same-order surds, compare the radicands. To compare with , square both.
For different orders, take the LCM of the orders and rewrite both as roots of that order — then compare radicands.
Rationalise BEFORE substituting numerical values.** Dividing by a small difference of two roots loses accuracy; adding does not.
And when the two radicands differ by , the conjugate leaves a denominator of — so the answer is just the conjugate itself.
A surd in lowest form has no square factor inside, no fraction inside, and the smallest possible order.
Simplify every surd BEFORE adding or subtracting. looks unlike until you write .
Only LIKE surds can be added — same order and same radicand.
** is NOT .** Check: and . But is true.
**The conjugate of is **, and their product is rational.
Multiply the numerator by the conjugate too — in a case like the numerator becomes a square, so expand it carefully.
Always check a rationalised answer numerically. One substitution catches a sign error instantly.
**To compare same-order surds, compare the radicands. To compare with , square both.
For different orders, take the LCM of the orders and rewrite both as roots of that order — then compare radicands.
Rationalise BEFORE substituting numerical values.** Dividing by a small difference of two roots loses accuracy; adding does not.
And when the two radicands differ by , the conjugate leaves a denominator of — so the answer is just the conjugate itself.
Did you know
Why subtracting two nearly equal numbers is the most dangerous step in arithmetic
There is a step in ordinary calculation that quietly destroys accuracy, and almost nobody is warned about it.
Suppose you know and to three decimal places — and . That is four significant figures each, which sounds comfortable.
Now subtract: .
Look at what happened to the significant figures. Each input had four. The answer has three, and the leading digits that agreed have cancelled out entirely. The and that you were confident about have vanished, and what remains is the difference between the parts you were least sure of.
That is the problem. The uncertainty in each root was in its last digit — about . That uncertainty is still there in the answer, but the answer is now only instead of , so the same absolute error is a much larger share of the result.
And then you divide by it. Dividing by multiplies everything, including the error, by about three. The final answer is out by roughly .
Rationalising avoids the whole thing. is an addition, and addition never cancels leading digits — the two uncertainties stay the same size relative to a result of . The answer comes out about five times closer to the truth.
So the mathematical identity and the numerical advice point the same way, which is unusual and worth noticing. Rationalising the denominator is not a tidying convention that happens to be harmless; it is the better way to compute.
This has a name outside school mathematics — the loss of significant figures when nearly equal quantities are subtracted — and it is the reason a formula that is algebraically correct can still be the wrong formula to put into a calculator. Two expressions can be exactly equal and one of them can be far more trustworthy to evaluate.
Which is a useful thing to carry forward. Algebraic rearrangement is not always cosmetic — sometimes it changes how much of your answer you can believe.
Suppose you know and to three decimal places — and . That is four significant figures each, which sounds comfortable.
Now subtract: .
Look at what happened to the significant figures. Each input had four. The answer has three, and the leading digits that agreed have cancelled out entirely. The and that you were confident about have vanished, and what remains is the difference between the parts you were least sure of.
That is the problem. The uncertainty in each root was in its last digit — about . That uncertainty is still there in the answer, but the answer is now only instead of , so the same absolute error is a much larger share of the result.
And then you divide by it. Dividing by multiplies everything, including the error, by about three. The final answer is out by roughly .
Rationalising avoids the whole thing. is an addition, and addition never cancels leading digits — the two uncertainties stay the same size relative to a result of . The answer comes out about five times closer to the truth.
So the mathematical identity and the numerical advice point the same way, which is unusual and worth noticing. Rationalising the denominator is not a tidying convention that happens to be harmless; it is the better way to compute.
This has a name outside school mathematics — the loss of significant figures when nearly equal quantities are subtracted — and it is the reason a formula that is algebraically correct can still be the wrong formula to put into a calculator. Two expressions can be exactly equal and one of them can be far more trustworthy to evaluate.
Which is a useful thing to carry forward. Algebraic rearrangement is not always cosmetic — sometimes it changes how much of your answer you can believe.
Exam relevance
Why does JEE Main keep asking you to rationalise?
Because a surd answer has to be in a standard form before it can be matched against the options, and rationalising is what puts it there.
This is the foundation for Class 11 Mathematics Complex Numbers and Quadratic Equations, Sequences and Series and Limits and Derivatives, examined in JEE Main. The conjugate method learnt here transfers directly: to simplify you multiply by the complex conjugate , and the denominator becomes — the identical trick with in place of . **Questions on expressing a complex number in the form are standard in JEE Main, and they are this section with one sign changed.
Surd roots of quadratics come out in conjugate pairs.** Class 11 shows that if a quadratic with rational coefficients has an irrational root , then is also a root. So conjugates stop being a computational trick and become a theorem, and questions asking for the other root of such a quadratic are answered from it.
Rationalising is the standard device for indeterminate limits. Class 11 Limits and Derivatives evaluates limits such as by multiplying top and bottom by the conjugate — which turns the numerator into and lets the cancel. This is one of the most reliably examined techniques in the limits chapter, and it is exactly the method on this page used on an algebraic rather than a numerical expression.
The loss-of-accuracy point is examined indirectly. Questions that ask for an answer "in simplest surd form" or "with a rational denominator" are enforcing this convention, and an answer left as will often not match any option even though it is correct.
Laws of indices generalise the order comparison. Class 11 treats as and the comparison method of this page as a use of the index laws — so "take the LCM of the orders" becomes "write both with a common fractional exponent". **Questions comparing , and appear in JEE Main, and the LCM method answers them all.
Surds appear throughout coordinate geometry and trigonometry.** Distances come out as surds from the distance formula, and the exact values of trigonometric ratios at , and degrees are surds — so and expressions such as must be rationalised to be recognised. Answers must be left in surd form, not converted to decimals.
What the questions look like. For board work, expect simplify a given surd, add, subtract, multiply or divide given surds, rationalise the denominator of a given expression, compare or arrange surds in order, and evaluate an expression given approximate values of the roots. The rationalising must be shown, not assumed. For JEE Main, expect complex-number conjugates, conjugate surd roots of a quadratic, rationalisation in limits, and index-law comparisons.
How board and competitive emphasis differ. A board paper rewards the worked rationalisation with the conjugate written out. A competitive paper assumes it and uses it as one step inside a limit, a complex number or a quadratic.
The single trap that costs the most marks. Writing . A square root distributes over a product and never over a sum — while , which are not remotely equal. **The defence is to notice that this is the same error as writing **, and both come from treating an operation as distributive when it is not. If you would not drop the from a square, do not drop it from a root — and a five-second numerical check on small numbers will expose the mistake every time.
This is the foundation for Class 11 Mathematics Complex Numbers and Quadratic Equations, Sequences and Series and Limits and Derivatives, examined in JEE Main. The conjugate method learnt here transfers directly: to simplify you multiply by the complex conjugate , and the denominator becomes — the identical trick with in place of . **Questions on expressing a complex number in the form are standard in JEE Main, and they are this section with one sign changed.
Surd roots of quadratics come out in conjugate pairs.** Class 11 shows that if a quadratic with rational coefficients has an irrational root , then is also a root. So conjugates stop being a computational trick and become a theorem, and questions asking for the other root of such a quadratic are answered from it.
Rationalising is the standard device for indeterminate limits. Class 11 Limits and Derivatives evaluates limits such as by multiplying top and bottom by the conjugate — which turns the numerator into and lets the cancel. This is one of the most reliably examined techniques in the limits chapter, and it is exactly the method on this page used on an algebraic rather than a numerical expression.
The loss-of-accuracy point is examined indirectly. Questions that ask for an answer "in simplest surd form" or "with a rational denominator" are enforcing this convention, and an answer left as will often not match any option even though it is correct.
Laws of indices generalise the order comparison. Class 11 treats as and the comparison method of this page as a use of the index laws — so "take the LCM of the orders" becomes "write both with a common fractional exponent". **Questions comparing , and appear in JEE Main, and the LCM method answers them all.
Surds appear throughout coordinate geometry and trigonometry.** Distances come out as surds from the distance formula, and the exact values of trigonometric ratios at , and degrees are surds — so and expressions such as must be rationalised to be recognised. Answers must be left in surd form, not converted to decimals.
What the questions look like. For board work, expect simplify a given surd, add, subtract, multiply or divide given surds, rationalise the denominator of a given expression, compare or arrange surds in order, and evaluate an expression given approximate values of the roots. The rationalising must be shown, not assumed. For JEE Main, expect complex-number conjugates, conjugate surd roots of a quadratic, rationalisation in limits, and index-law comparisons.
How board and competitive emphasis differ. A board paper rewards the worked rationalisation with the conjugate written out. A competitive paper assumes it and uses it as one step inside a limit, a complex number or a quadratic.
The single trap that costs the most marks. Writing . A square root distributes over a product and never over a sum — while , which are not remotely equal. **The defence is to notice that this is the same error as writing **, and both come from treating an operation as distributive when it is not. If you would not drop the from a square, do not drop it from a root — and a five-second numerical check on small numbers will expose the mistake every time.
Key takeaways
Surds and rationalising: quick revision
- A surd is an irrational root of a rational number. In , is the order and the radicand.
- Lowest form: no perfect-square factor inside, no fraction inside, and the smallest order.
- Take out the LARGEST square factor: , , , , .
- Prime-factorise if unsure — , and each pair of equal primes comes out.
- Clear a fraction from under the root: , and squaring the answer gives back.
- Like surds have the same order and radicand, and only like surds can be added or subtracted.
- . .
- Multiplication: ; .
- . .
- Division: ; .
- ** — but is true. A root distributes over a product, never a sum.
- Conjugate** of is , and , which is rational.
- . . . .
- **When the radicands differ by ** the denominator becomes , so .
- Same-order comparison: exactly when . With coefficients, square both: , so is larger.
- Different orders — take the LCM: and , so . And since .
- Rationalise BEFORE substituting: direct gives , rationalised gives , and the true value is — the rationalised route is about five times more accurate from the same data.
- .
- Subtracting two nearly equal numbers destroys significant figures, and dividing by the small result magnifies what is left — which is the real reason the convention exists.
Take any expression with a two-term surd denominator, rationalise it, and then evaluate both forms from three-figure roots — the gap between the two answers is the whole argument of this chapter.
- Lowest form: no perfect-square factor inside, no fraction inside, and the smallest order.
- Take out the LARGEST square factor: , , , , .
- Prime-factorise if unsure — , and each pair of equal primes comes out.
- Clear a fraction from under the root: , and squaring the answer gives back.
- Like surds have the same order and radicand, and only like surds can be added or subtracted.
- . .
- Multiplication: ; .
- . .
- Division: ; .
- ** — but is true. A root distributes over a product, never a sum.
- Conjugate** of is , and , which is rational.
- . . . .
- **When the radicands differ by ** the denominator becomes , so .
- Same-order comparison: exactly when . With coefficients, square both: , so is larger.
- Different orders — take the LCM: and , so . And since .
- Rationalise BEFORE substituting: direct gives , rationalised gives , and the true value is — the rationalised route is about five times more accurate from the same data.
- .
- Subtracting two nearly equal numbers destroys significant figures, and dividing by the small result magnifies what is left — which is the real reason the convention exists.
Take any expression with a two-term surd denominator, rationalise it, and then evaluate both forms from three-figure roots — the gap between the two answers is the whole argument of this chapter.