A Cube Keeps the Sign of Its Number but a Square Never Does
Learn to test whether a number is a perfect cube, cube negative numbers, fractions and decimals, find cube roots by prime factorisation and from the last digit, and find the smallest multiplier or divisor that makes a perfect cube.
Why can a cube be negative when a square never can?
Because a cube multiplies the number by itself an odd number of times.
In the square the two minus signs cancel. In the cube there is a third minus left over with nothing to pair with, so the answer stays negative.
So a cube keeps the sign of the number it came from, and this means every real number has a cube root — including the negative ones, which have no square roots at all. This page covers the ICSE Class 8 Mathematics chapter on cubes and cube roots.
In the square the two minus signs cancel. In the cube there is a third minus left over with nothing to pair with, so the answer stays negative.
So a cube keeps the sign of the number it came from, and this means every real number has a cube root — including the negative ones, which have no square roots at all. This page covers the ICSE Class 8 Mathematics chapter on cubes and cube roots.
How do you tell whether a number is a perfect cube?
A perfect cube is a number obtained by multiplying an integer by itself three times. The first few are
The test: break the number into prime factors and check that every exponent is a multiple of three. Equivalently, group the prime factors in threes and check that none is left over.
Worked example 1. Is a perfect cube?
Dividing repeatedly:
The exponents are and . Both are multiples of three, so is a perfect cube. Dividing each exponent by three gives the root:
Checking: . Correct.
Worked example 2. Is a perfect cube?
The exponent of is **, which is not** a multiple of three, so is not a perfect cube. One group of three twos cannot be formed.
Worked example 3. Is a perfect cube?
Both exponents are , so yes, and the root is . Checking: . Correct.
Comparing this with squares. For a square, every exponent must be a multiple of two; for a cube, a multiple of three. The whole test is the same idea with a different divisor, which is why the two chapters look so alike.
A useful consequence. Since the exponents must be multiples of three, perfect cubes are far rarer than perfect squares. Between and there are perfect squares but only **** perfect cubes — which is worth knowing when a question asks you to guess whether a large number is one.
The test: break the number into prime factors and check that every exponent is a multiple of three. Equivalently, group the prime factors in threes and check that none is left over.
Worked example 1. Is a perfect cube?
Dividing repeatedly:
The exponents are and . Both are multiples of three, so is a perfect cube. Dividing each exponent by three gives the root:
Checking: . Correct.
Worked example 2. Is a perfect cube?
The exponent of is **, which is not** a multiple of three, so is not a perfect cube. One group of three twos cannot be formed.
Worked example 3. Is a perfect cube?
Both exponents are , so yes, and the root is . Checking: . Correct.
Comparing this with squares. For a square, every exponent must be a multiple of two; for a cube, a multiple of three. The whole test is the same idea with a different divisor, which is why the two chapters look so alike.
A useful consequence. Since the exponents must be multiples of three, perfect cubes are far rarer than perfect squares. Between and there are perfect squares but only **** perfect cubes — which is worth knowing when a question asks you to guess whether a large number is one.
Formula
How do you cube a negative number, a fraction or a decimal?
Cube each part, and let the sign look after itself.
A negative number.
The sign is retained — a negative number's cube is negative, and a positive number's cube is positive. Contrast this with a square, which is always positive.
A fraction — cube the numerator and the denominator separately.
A decimal — count the decimal places carefully. A number with decimal places has a cube with **** decimal places.
Here has one decimal place, so the cube has three. Working it as a fraction confirms it: .
Again one place in, three places out — and checks it.
The decimal-place rule is the commonest source of error. A student who reasons *, so * has lost a place. Converting the decimal to a fraction over a power of ten removes the guesswork entirely, and is worth doing whenever the number of places matters.
A mixed number must be converted first. is not . Convert to an improper fraction:
And the sign rule in one line. An odd power keeps the sign; an even power destroys it. So while — which is the same rule met in the chapter on exponents, appearing here in its most useful special case.
A negative number.
The sign is retained — a negative number's cube is negative, and a positive number's cube is positive. Contrast this with a square, which is always positive.
A fraction — cube the numerator and the denominator separately.
A decimal — count the decimal places carefully. A number with decimal places has a cube with **** decimal places.
Here has one decimal place, so the cube has three. Working it as a fraction confirms it: .
Again one place in, three places out — and checks it.
The decimal-place rule is the commonest source of error. A student who reasons *, so * has lost a place. Converting the decimal to a fraction over a power of ten removes the guesswork entirely, and is worth doing whenever the number of places matters.
A mixed number must be converted first. is not . Convert to an improper fraction:
And the sign rule in one line. An odd power keeps the sign; an even power destroys it. So while — which is the same rule met in the chapter on exponents, appearing here in its most useful special case.
How do you find a cube root without a calculator?
Two ways: prime factorisation for exact work, and the last-digit method for cubes up to six digits.
By prime factorisation, group the primes in threes and take one factor from each group.
Worked example 1. Find .
Checking: . Correct.
Worked example 2. Find .
Dividing each exponent by three:
Checking: . Correct.
Worked example 3 — negatives and fractions.
since , and . The cube root of a negative number is simply negative — no special case needed.
The last-digit method, which is much faster for a perfect cube of up to six digits. It relies on the fact that the last digit of a cube fixes the last digit of its root:
- a cube ending in has a root ending in
- ending in gives a root ending in
- ending in gives a root ending in
- ending in gives a root ending in
- ending in gives , ending in gives , ending in gives
- ending in gives a root ending in
- ending in gives a root ending in
- ending in gives a root ending in
Worked example 4. Find .
- The number ends in **, so the root ends in .
- Strike off the last three digits**, leaving . Since , the tens digit is ****.
- So the root is .
Checking: and . Correct.
Worked example 5. Find .
- It ends in **, so the root ends in **.
- Striking off the last three digits leaves , and , so the tens digit is ****.
- The root is , and confirms it.
The limitation of the last-digit method. It works only on a perfect cube. Given a number that is not one, it will still produce an answer — and that answer will be wrong. So use it for speed and confirm by cubing the result, which takes one line.
By prime factorisation, group the primes in threes and take one factor from each group.
Worked example 1. Find .
Checking: . Correct.
Worked example 2. Find .
Dividing each exponent by three:
Checking: . Correct.
Worked example 3 — negatives and fractions.
since , and . The cube root of a negative number is simply negative — no special case needed.
The last-digit method, which is much faster for a perfect cube of up to six digits. It relies on the fact that the last digit of a cube fixes the last digit of its root:
- a cube ending in has a root ending in
- ending in gives a root ending in
- ending in gives a root ending in
- ending in gives a root ending in
- ending in gives , ending in gives , ending in gives
- ending in gives a root ending in
- ending in gives a root ending in
- ending in gives a root ending in
Worked example 4. Find .
- The number ends in **, so the root ends in .
- Strike off the last three digits**, leaving . Since , the tens digit is ****.
- So the root is .
Checking: and . Correct.
Worked example 5. Find .
- It ends in **, so the root ends in **.
- Striking off the last three digits leaves , and , so the tens digit is ****.
- The root is , and confirms it.
The limitation of the last-digit method. It works only on a perfect cube. Given a number that is not one, it will still produce an answer — and that answer will be wrong. So use it for speed and confirm by cubing the result, which takes one line.
How do you make a number into a perfect cube?
Factorise it, find the primes whose exponents are not multiples of three, and supply or remove exactly what is missing.
Worked example 1 — multiplying. Find the smallest number by which must be multiplied to make it a perfect cube.
The exponent of is , which is fine. The exponent of is **, so it needs one more ** to reach .
So multiply by ****:
Checking: . Correct.
Worked example 2 — dividing. Find the smallest number by which must be divided to make it a perfect cube.
The exponents of and are fine. The exponent of is **, and the cheaper fix is to remove** that single rather than supply two more.
So divide by ****:
Checking: . Correct.
How to decide between multiplying and dividing. A question usually tells you which. If it does not, compare the work: an exponent of needs two more factors to multiply up but only one to divide out, while an exponent of needs one to multiply up and two to divide out. Whichever is smaller is the answer.
Worked example 3 — a word problem. The volume of a cube is . Find its edge and its total surface area.
The volume of a cube is , so
The surface area is times the area of one face:
Checking by the last-digit method: ends in **, so the root ends in **; striking the last three digits leaves , and , so the tens digit is — giving , as found.
Why this problem type is so common. A cube is the one solid whose volume is a single length cubed, so a cube root turns a volume straight back into a length. Note the units: in gives out, and the surface area is then back in — three different powers of the same unit in one question.
A cubes-and-squares boundary case. is both a perfect square () and a perfect cube (), because and is a multiple of both and . The next such number is , being and — so numbers of the form are exactly the ones that are both.
Worked example 1 — multiplying. Find the smallest number by which must be multiplied to make it a perfect cube.
The exponent of is , which is fine. The exponent of is **, so it needs one more ** to reach .
So multiply by ****:
Checking: . Correct.
Worked example 2 — dividing. Find the smallest number by which must be divided to make it a perfect cube.
The exponents of and are fine. The exponent of is **, and the cheaper fix is to remove** that single rather than supply two more.
So divide by ****:
Checking: . Correct.
How to decide between multiplying and dividing. A question usually tells you which. If it does not, compare the work: an exponent of needs two more factors to multiply up but only one to divide out, while an exponent of needs one to multiply up and two to divide out. Whichever is smaller is the answer.
Worked example 3 — a word problem. The volume of a cube is . Find its edge and its total surface area.
The volume of a cube is , so
The surface area is times the area of one face:
Checking by the last-digit method: ends in **, so the root ends in **; striking the last three digits leaves , and , so the tens digit is — giving , as found.
Why this problem type is so common. A cube is the one solid whose volume is a single length cubed, so a cube root turns a volume straight back into a length. Note the units: in gives out, and the surface area is then back in — three different powers of the same unit in one question.
A cubes-and-squares boundary case. is both a perfect square () and a perfect cube (), because and is a multiple of both and . The next such number is , being and — so numbers of the form are exactly the ones that are both.
Exam tip
Exam tip: group the primes in threes, not pairs
For a cube root, group the prime factors in threes and divide every exponent by three. Halving the exponents is the square-root rule and is the error to guard against when both chapters are fresh.
A number is a perfect cube exactly when every prime exponent is a multiple of three.
Remember the sign rule: a cube keeps the sign, so and . Every number has a cube root, including negatives.
For a decimal, a number with decimal places cubes to **** places: , not . Converting to a fraction over a power of ten removes the doubt.
Convert a mixed number to an improper fraction first: .
For the last-digit method, state both steps: the units digit from the last digit of the cube, and the tens digit from the part left after striking off the last three digits. It works only on a perfect cube, so check by cubing your answer.
When finding the smallest multiplier or divisor, write the factorisation with exponents and name the prime that is short — that line is where the method mark is.
And in a volume problem, track the units: .
A number is a perfect cube exactly when every prime exponent is a multiple of three.
Remember the sign rule: a cube keeps the sign, so and . Every number has a cube root, including negatives.
For a decimal, a number with decimal places cubes to **** places: , not . Converting to a fraction over a power of ten removes the doubt.
Convert a mixed number to an improper fraction first: .
For the last-digit method, state both steps: the units digit from the last digit of the cube, and the tens digit from the part left after striking off the last three digits. It works only on a perfect cube, so check by cubing your answer.
When finding the smallest multiplier or divisor, write the factorisation with exponents and name the prime that is short — that line is where the method mark is.
And in a volume problem, track the units: .
Did you know
Why is every cube the sum of a run of consecutive odd numbers?
Pick a cube and it can always be written as a sum of consecutive odd numbers:
The pattern is exact. The first cube takes one odd number, the second takes the next two, the third the next three, and so on — and the odd numbers are used up in order without a single gap or repeat.
So the whole sequence of odd numbers, sliced into groups of sizes , produces the cubes in order. Checking the fourth group: .
It pairs neatly with the square rule from the previous chapter, where the sum of the **first ** odd numbers is . The same supply of odd numbers, added up from the start, gives the squares; grouped into runs of increasing length, it gives the cubes. Two of the most useful patterns in arithmetic come out of one sequence, read two different ways.
The pattern is exact. The first cube takes one odd number, the second takes the next two, the third the next three, and so on — and the odd numbers are used up in order without a single gap or repeat.
So the whole sequence of odd numbers, sliced into groups of sizes , produces the cubes in order. Checking the fourth group: .
It pairs neatly with the square rule from the previous chapter, where the sum of the **first ** odd numbers is . The same supply of odd numbers, added up from the start, gives the squares; grouped into runs of increasing length, it gives the cubes. Two of the most useful patterns in arithmetic come out of one sequence, read two different ways.
Key takeaways
Cubes and cube roots: quick revision
- A perfect cube is an integer cubed: .
- Test: every prime exponent must be a multiple of three. is a cube; is not, because is not a multiple of three.
- Cube root by factorisation divides every exponent by three: , , , .
- A cube keeps the sign: and . Every number has a cube root, unlike square roots.
- Fractions: cube each part — , , .
- Decimals: places in gives **** places out — and . Convert a mixed number first: .
- Last-digit method for a perfect cube: the units digit of the root comes from the last digit of the cube (, , , , , , , , , ), and the tens digit from the part left after striking off the last three digits. So and . It works only on a perfect cube, so check by cubing.
- Smallest multiplier: needs one more , so .
- Smallest divisor: has a spare , so .
- Volume problem: a cube of volume has edge and surface area .
- Numbers of the form — such as and — are both perfect squares and perfect cubes.
Try testing a dozen numbers for being perfect cubes from their factorisations, and check each cube root by the last-digit method as well — two independent routes agreeing is the surest confirmation.
- Test: every prime exponent must be a multiple of three. is a cube; is not, because is not a multiple of three.
- Cube root by factorisation divides every exponent by three: , , , .
- A cube keeps the sign: and . Every number has a cube root, unlike square roots.
- Fractions: cube each part — , , .
- Decimals: places in gives **** places out — and . Convert a mixed number first: .
- Last-digit method for a perfect cube: the units digit of the root comes from the last digit of the cube (, , , , , , , , , ), and the tens digit from the part left after striking off the last three digits. So and . It works only on a perfect cube, so check by cubing.
- Smallest multiplier: needs one more , so .
- Smallest divisor: has a spare , so .
- Volume problem: a cube of volume has edge and surface area .
- Numbers of the form — such as and — are both perfect squares and perfect cubes.
Try testing a dozen numbers for being perfect cubes from their factorisations, and check each cube root by the last-digit method as well — two independent routes agreeing is the surest confirmation.