If the Bases Match, You Can Throw the Bases Away
Learn to solve exponential equations by making the bases the same, write very large and very small numbers in standard form and back again, evaluate expressions with rational bases and reciprocals, and compare quantities with exponents.
How do you solve an equation with the unknown in the exponent?
Make both sides the same base, and then the exponents must be equal.
To solve , notice that . The equation becomes
and since the bases are identical, the exponents must match, giving . The bases have done their job and can be discarded.
This works because a positive base raised to two different powers cannot give the same answer twice — so matching values force matching exponents. This page covers the second part of the ICSE Class 8 Mathematics chapter on exponents: solving for an unknown power, standard form, and using exponents on real measurements.
To solve , notice that . The equation becomes
and since the bases are identical, the exponents must match, giving . The bases have done their job and can be discarded.
This works because a positive base raised to two different powers cannot give the same answer twice — so matching values force matching exponents. This page covers the second part of the ICSE Class 8 Mathematics chapter on exponents: solving for an unknown power, standard form, and using exponents on real measurements.
How do you find the unknown in an exponential equation?
There are two situations, and which one you are in depends on where the unknown sits.
Case 1 — the unknown is the exponent. Express both sides with the same base and equate the exponents.
Worked example 1. Solve .
Worked example 2. Solve .
Worked example 3. Solve .
Worked example 4 — bases that need converting. Solve .
Neither side is a power of the other, but both are powers of :
So an exponent need not be a whole number. Checking: means the square root of cubed, which is . Correct.
Worked example 5 — a rational base. Solve .
The right-hand side is the reciprocal cubed:
so . Notice the answer is negative, because the right-hand side is greater than while the base is a proper fraction.
Case 2 — the unknown is the base. Express both sides with the same exponent and equate the bases.
Worked example 6. Solve .
How to spot which case you are in. If the letter is upstairs, make the bases match. If the letter is downstairs, make the exponents match. And the first step in either case is the same: express the plain number as a power, which is why knowing the small powers of , and by heart makes this section quick.
A boundary case worth knowing. The method needs the base to be a positive number other than . With base it fails, because for every , so has infinitely many solutions and tells you nothing about .
Case 1 — the unknown is the exponent. Express both sides with the same base and equate the exponents.
Worked example 1. Solve .
Worked example 2. Solve .
Worked example 3. Solve .
Worked example 4 — bases that need converting. Solve .
Neither side is a power of the other, but both are powers of :
So an exponent need not be a whole number. Checking: means the square root of cubed, which is . Correct.
Worked example 5 — a rational base. Solve .
The right-hand side is the reciprocal cubed:
so . Notice the answer is negative, because the right-hand side is greater than while the base is a proper fraction.
Case 2 — the unknown is the base. Express both sides with the same exponent and equate the bases.
Worked example 6. Solve .
How to spot which case you are in. If the letter is upstairs, make the bases match. If the letter is downstairs, make the exponents match. And the first step in either case is the same: express the plain number as a power, which is why knowing the small powers of , and by heart makes this section quick.
A boundary case worth knowing. The method needs the base to be a positive number other than . With base it fails, because for every , so has infinitely many solutions and tells you nothing about .
Formula
How do you write a number in standard form and convert it back?
Standard form, also called scientific notation, writes a number as
where and is an integer.
The condition on is what makes the form unique: exactly one digit before the decimal point, and that digit not zero.
Converting a large number. Move the decimal point left until one non-zero digit remains before it, and let be the number of places moved, taken as positive.
The point moved eight places left, so . Checking: restores all eight places. Correct.
Converting a small number. Move the decimal point right until one non-zero digit stands before it, and let be the number of places moved, taken as negative.
The point moved eleven places right, so .
Converting back to usual form. A positive moves the point right; a negative moves it left, filling with zeros.
The rule in one line. A number **greater than has a positive exponent; a number less than has a negative one. So if a tiny number comes out with a positive power, or a huge one with a negative power, the sign has been reversed — which is the error to check for first.
Why the form is worth the trouble.** Written in full, is easy to miscount and easy to mistype, and comparing it with by eye is nearly impossible. Written as and , the difference is immediate.
And it makes arithmetic simple. Multiply the parts and add the exponents; divide the parts and subtract them — the laws of the previous part, applied to measurements:
where and is an integer.
The condition on is what makes the form unique: exactly one digit before the decimal point, and that digit not zero.
Converting a large number. Move the decimal point left until one non-zero digit remains before it, and let be the number of places moved, taken as positive.
The point moved eight places left, so . Checking: restores all eight places. Correct.
Converting a small number. Move the decimal point right until one non-zero digit stands before it, and let be the number of places moved, taken as negative.
The point moved eleven places right, so .
Converting back to usual form. A positive moves the point right; a negative moves it left, filling with zeros.
The rule in one line. A number **greater than has a positive exponent; a number less than has a negative one. So if a tiny number comes out with a positive power, or a huge one with a negative power, the sign has been reversed — which is the error to check for first.
Why the form is worth the trouble.** Written in full, is easy to miscount and easy to mistype, and comparing it with by eye is nearly impossible. Written as and , the difference is immediate.
And it makes arithmetic simple. Multiply the parts and add the exponents; divide the parts and subtract them — the laws of the previous part, applied to measurements:
How do you evaluate an expression with rational bases and reciprocals?
Clear each negative exponent by flipping its fraction, work out the individual powers, and only then combine.
Worked example 1. Evaluate .
Flip each fraction and square it:
Now substitute:
Worked example 2 — a power of a power. Evaluate .
Multiply the exponents:
Worked example 3 — mixing the laws. Evaluate .
Same base throughout, so combine the exponents in one step:
Worked example 4 — with a zero exponent hidden in it. Evaluate .
A question of this shape is testing whether you spot that the answer is without computing twice.
The order of operations still applies. In example 1 the bracket was evaluated before the division, which is why the was done first. Exponents do not override BODMAS; they take their place within it, immediately after brackets.
The check worth making on every answer. Ask whether the size is plausible. must be smaller than , since repeatedly multiplying by a proper fraction shrinks a number — and is indeed much smaller. Conversely must be larger than , and it is.
Worked example 1. Evaluate .
Flip each fraction and square it:
Now substitute:
Worked example 2 — a power of a power. Evaluate .
Multiply the exponents:
Worked example 3 — mixing the laws. Evaluate .
Same base throughout, so combine the exponents in one step:
Worked example 4 — with a zero exponent hidden in it. Evaluate .
A question of this shape is testing whether you spot that the answer is without computing twice.
The order of operations still applies. In example 1 the bracket was evaluated before the division, which is why the was done first. Exponents do not override BODMAS; they take their place within it, immediately after brackets.
The check worth making on every answer. Ask whether the size is plausible. must be smaller than , since repeatedly multiplying by a proper fraction shrinks a number — and is indeed much smaller. Conversely must be larger than , and it is.
How are exponents used to compare real measurements?
By writing both quantities in standard form and dividing, so the comparison reduces to subtracting exponents.
Worked example 1 — how many times larger. How many times larger is than ?
Divide, taking the parts and the powers separately:
So the first is three lakh times the second — a comparison that would be laborious with the numbers written out in full.
Worked example 2 — adding many small quantities. A sheet of paper is about thick. How thick is a ream of sheets?
which is , or . A sensible answer for a ream of paper, which is how you know the powers were handled correctly.
Worked example 3 — comparing two powers directly. Which is greater, or ?
So is greater, though only just. This is a useful pair to know, because it means a thousand and are nearly interchangeable as rough figures.
Worked example 4 — a length comparison. A quantity measures and another . How many times longer is the first?
Fifty times longer. Note the double negative in the exponent subtraction: , and getting that sign wrong is the usual source of error in this kind of question.
Why standard form is worth insisting on here. Every one of these calculations became a subtraction of two small integers plus an easy division. Done with the numbers written out, the same problems would be exercises in counting zeros — which is slow and unreliable.
A final tidying rule. An answer should be put back into standard form if it is not already there. In example 4 the intermediate has , which breaks the condition , so it is rewritten as or simply .
Worked example 1 — how many times larger. How many times larger is than ?
Divide, taking the parts and the powers separately:
So the first is three lakh times the second — a comparison that would be laborious with the numbers written out in full.
Worked example 2 — adding many small quantities. A sheet of paper is about thick. How thick is a ream of sheets?
which is , or . A sensible answer for a ream of paper, which is how you know the powers were handled correctly.
Worked example 3 — comparing two powers directly. Which is greater, or ?
So is greater, though only just. This is a useful pair to know, because it means a thousand and are nearly interchangeable as rough figures.
Worked example 4 — a length comparison. A quantity measures and another . How many times longer is the first?
Fifty times longer. Note the double negative in the exponent subtraction: , and getting that sign wrong is the usual source of error in this kind of question.
Why standard form is worth insisting on here. Every one of these calculations became a subtraction of two small integers plus an easy division. Done with the numbers written out, the same problems would be exercises in counting zeros — which is slow and unreliable.
A final tidying rule. An answer should be put back into standard form if it is not already there. In example 4 the intermediate has , which breaks the condition , so it is rewritten as or simply .
Exam tip
Exam tip: express the plain number as a power first
The first move in any exponential equation is to write the ordinary number as a power of the same base: , , , . Learn the small powers of , and so this is instant.
Decide which case you are in: the unknown upstairs means equate exponents; the unknown downstairs means equate bases.
When the bases differ, convert both to a common base — becomes .
In standard form check that — exactly one non-zero digit before the decimal point. An answer of or is not yet in standard form.
Get the sign of the exponent right: a number greater than 1 takes a positive power, one less than 1 a negative power. Check this before anything else.
Count the decimal places carefully and state the count. needs eleven places, giving .
When dividing standard-form numbers, subtract the exponents and watch the double negative: .
Simplify with the exponent laws before evaluating — is , and there is no need to compute at all.
And carry the units through a measurement problem: .
Decide which case you are in: the unknown upstairs means equate exponents; the unknown downstairs means equate bases.
When the bases differ, convert both to a common base — becomes .
In standard form check that — exactly one non-zero digit before the decimal point. An answer of or is not yet in standard form.
Get the sign of the exponent right: a number greater than 1 takes a positive power, one less than 1 a negative power. Check this before anything else.
Count the decimal places carefully and state the count. needs eleven places, giving .
When dividing standard-form numbers, subtract the exponents and watch the double negative: .
Simplify with the exponent laws before evaluating — is , and there is no need to compute at all.
And carry the units through a measurement problem: .
Did you know
Why do computer memory sizes go up in awkward numbers?
Memory and storage come in sizes like , , and , never in round hundreds or thousands. Every one of those is a power of two.
The reason is that a computer stores everything as a pattern of two states, so the number of distinct patterns available from switches is . Sizes that are powers of two use every available pattern with none wasted, and any other size leaves some unusable.
What makes it look almost tidy is the near-coincidence from the worked example above: , which is only a little more than . So a quantity of is loosely called a thousand, is loosely called a million, and the everyday names line up with the powers of two closely enough to be convenient.
That small gap is why a storage device sold as holding a round number of units always seems to show slightly less when you look at it — the two counting systems were never exactly equal, only close.
The reason is that a computer stores everything as a pattern of two states, so the number of distinct patterns available from switches is . Sizes that are powers of two use every available pattern with none wasted, and any other size leaves some unusable.
What makes it look almost tidy is the near-coincidence from the worked example above: , which is only a little more than . So a quantity of is loosely called a thousand, is loosely called a million, and the everyday names line up with the powers of two closely enough to be convenient.
That small gap is why a storage device sold as holding a round number of units always seems to show slightly less when you look at it — the two counting systems were never exactly equal, only close.
Key takeaways
Exponential equations and standard form: quick revision
- Unknown in the exponent: make the bases the same and equate the exponents. gives ; gives ; gives .
- When the bases differ, convert both: becomes , so .
- With a rational base: , so .
- Unknown in the base: make the exponents the same and equate the bases. gives .
- The method fails for base , since for every .
- Standard form: with and an integer.
- Large numbers take a positive exponent: . Small numbers take a negative one: .
- Back to usual form: and .
- Evaluating with rational bases — flip the negative exponents first: ; ; ; .
- BODMAS still applies — brackets before exponents in an expression.
- Comparing quantities: ; sheets of paper make ; just exceeds ; and .
- Watch the double negative when subtracting exponents, and return the final answer to standard form.
Practise a mixed set of exponential equations and standard-form conversions, checking the sign of every exponent against whether the number is above or below .
- When the bases differ, convert both: becomes , so .
- With a rational base: , so .
- Unknown in the base: make the exponents the same and equate the bases. gives .
- The method fails for base , since for every .
- Standard form: with and an integer.
- Large numbers take a positive exponent: . Small numbers take a negative one: .
- Back to usual form: and .
- Evaluating with rational bases — flip the negative exponents first: ; ; ; .
- BODMAS still applies — brackets before exponents in an expression.
- Comparing quantities: ; sheets of paper make ; just exceeds ; and .
- Watch the double negative when subtracting exponents, and return the final answer to standard form.
Practise a mixed set of exponential equations and standard-form conversions, checking the sign of every exponent against whether the number is above or below .