Some numbers are special. Multiply 12 by itself and you get 144. We say 144 is a perfect square and 12 is its square root. Do the same with three copies, and you get a cube.
These questions come up in almost every maths paper. Most of them can be done in under a minute. This article shows the formulas, the shortcuts and five solved examples. Each example says why the method works.
- Square root: the number that, times itself, gives the original.
- Cube root: the number that, used three times in a product, gives the original.
- Perfect square: every prime in its factors has an even power.
- Perfect cube: every prime has a power that is a multiple of 3.
- Unit digit: a square ends only in 0, 1, 4, 5, 6 or 9.
- Decimals: shift the decimal point by an even number of places for squares and by a multiple of 3 for cubes.
What are squares, cubes and their roots?
What is a square root?
The square root of x is the number that gives x when multiplied by itself. We write it as √x.
So √144 = 12, because 12 × 12 = 144. And √6400 = 80, because 80 × 80 = 6400.
Sources: NCERT Class 8 Maths: Squares and Square Roots, perfect squares; Practice question: square root of 144400.
What is a cube root?
The cube root of x is the number whose cube is x. We write it as ∛x.
So ∛1728 = 12, because 12 × 12 × 12 = 1728. And ∛0.125 = 0.5.
Sources: NCERT Class 8 Maths: Cubes and Cube Roots, perfect cubes; Practice question: cube root of 85184.
What is a perfect square, and what is a perfect cube?
A perfect square is a whole number that is the square of another whole number. The numbers 1, 4, 9, 16 and 25 are examples.
A perfect cube is the cube of a whole number. The numbers 1, 8, 27, 64 and 125 are examples.
- Squares from 1 to 15: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225.
- Cubes from 1 to 10: 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000.
Sources: NCERT Class 8 Maths: Squares and Square Roots, unit digits of squares; NCERT Class 8 Maths: Cubes and Cube Roots, perfect cubes.
How can the last digit tell you if a number is a square?
Look at the unit digit. A perfect square ends only in 0, 1, 4, 5, 6 or 9.
It can never end in 2, 3, 7 or 8. So 4587 and 2038 are not perfect squares.
Sources: NCERT Class 8 Maths: Squares and Square Roots, unit digits of squares; Practice question: unit digit of a square root.
What is the unit digit of a square root?
Use the unit digit of the square. If it is 1, the root ends in 1 or 9. If it is 4, the root ends in 2 or 8.
If it is 5, the root ends in 5. If it is 6, the root ends in 4 or 6. If it is 9, the root ends in 3 or 7. If it is 0, the root ends in 0.
Sources: NCERT Class 8 Maths: Squares and Square Roots, unit digits of squares; Practice question: unit digit of root of 29241.
What is the unit digit of a cube root?
Cubes are kind to us. Each last digit of a cube points to exactly one last digit of the root.
A cube ending in 8 has a root ending in 2. A cube ending in 2 has a root ending in 8. Digits 0, 1, 4, 5, 6 and 9 map to themselves. Digits 3 and 7 swap.
Sources: NCERT Class 8 Maths: Cubes and Cube Roots, perfect cubes.
Which methods and shortcuts find the roots quickly?
How do you find a square root by prime factorisation?
Break the number into prime factors. Group equal primes in pairs. Take one prime from each pair, and multiply them.
For 3136, the factors are 2 × 2 × 2 × 2 × 2 × 2 × 7 × 7. The pairs give 2 × 2 × 2 × 7 = 56.
Sources: NCERT Class 8 Maths: Squares and Square Roots, finding roots by prime factorisation.
How do you find a cube root by prime factorisation?
The method is the same, but you make triplets instead of pairs. Take one prime from each triplet.
For 5832, the factors are 2 × 2 × 2 × 3 × 3 × 3 × 3 × 3 × 3. The triplets give 2 × 3 × 3 = 18.
Sources: NCERT Class 8 Maths: Cubes and Cube Roots, roots by prime factorisation.
How do you find the square root of a decimal?
Remove the decimal point, find the root of the whole number, then put the point back. Count the places carefully.
The decimal part must have an even number of places. So √0.2304 = 0.48, because 48 × 48 = 2304 and 4 places become 2 places.
Sources: NCERT Class 8 Maths: Squares and Square Roots, roots of decimals; Practice question: square root of 0.2304.
How does moving the decimal point change a root?
For squares, two places in the number means one place in the root. So √6.25 = 2.5 and √0.0625 = 0.25.
For cubes, three places in the number means one place in the root. So ∛0.027 = 0.3 and ∛0.000027 = 0.03.
Sources: NCERT Class 8 Maths: Squares and Square Roots, roots of decimals; Practice question: sum of roots using √625 = 25.
How many perfect squares lie between two numbers?
Find the smallest and the largest whole numbers whose squares lie inside the range. Count the whole numbers between them.
Between 80 and 280, the squares run from 9² = 81 to 16² = 256. So there are 16 − 9 + 1 = 8 of them.
Sources: NCERT Class 8 Maths: Squares and Square Roots, numbers between squares; Practice question: perfect squares between 80 and 280.
How do you make a number a perfect square or a perfect cube?
Factorise the number. For a square, check each prime for an odd power and multiply by the missing prime. For a cube, fix any power that is not a multiple of 3.
Dividing works the other way. Divide out the extra primes that have no partner.
- Multiply to a square: 288 = 2⁵ × 3², so multiply by 2.
- Multiply to a cube: 250 = 2 × 5³, so multiply by 2² = 4.
- Divide to a square: 588 = 2² × 3 × 7², so divide by 3.
Sources: NCERT Class 8 Maths: Cubes and Cube Roots, making a number a perfect cube; Practice question: least number to multiply 288.
Check 1
A perfect square can end in the digit 9.
Check 2
A perfect square can end in the digit 8.
Check 3
The cube root of 0.064 is 0.4.
Check 4
The square root of 0.49 is 0.07.
How do you solve five typical problems step by step?
Example 1: Find the square root of 3136.
Step 1. Factorise: 3136 = 2⁶ × 7².
Step 2. Halve each power: 2³ × 7¹.
Step 3. Multiply: 8 × 7 = 56.
Check. 56 × 56 = 3136.
Why this works. A square is a product of a number with itself. So every prime appears an even number of times, and half of them belong to the root.
Example 2: Find the cube root of 5832.
Step 1. Factorise: 5832 = 2³ × 3⁶.
Step 2. Divide each power by 3: 2¹ × 3².
Step 3. Multiply: 2 × 9 = 18.
Check. 18 × 18 = 324, and 324 × 18 = 5832.
Why this works. A cube is a product of three equal numbers. So each prime appears in groups of three, and one from each group belongs to the root.
Sources: Practice question: cube root of 5832.
Example 3: By what least number should 180 be multiplied to get a perfect square?
Step 1. Factorise: 180 = 2² × 3² × 5.
Step 2. The primes 2 and 3 are paired. The prime 5 is alone.
Step 3. Multiply by 5. The product is 900.
Check. 900 = 30 × 30.
Why this works. Only the unpaired prime needs a partner. Anything more would be a waste.
Sources: Practice question: least number to make a perfect square.
Example 4: By what least number should 972 be divided to get a perfect square?
Step 1. Factorise: 972 = 2² × 3⁵.
Step 2. The power 5 of the prime 3 is odd. One extra 3 has no partner.
Step 3. Divide by 3. The result is 324.
Check. 324 = 18 × 18.
Why this works. Removing the lone factor leaves only pairs behind.
Example 5: Given √529 = 23, find √5.29 + √0.0529.
Step 1. 5.29 is 529 divided by 100. So √5.29 = 23 ÷ 10 = 2.3.
Step 2. 0.0529 is 529 divided by 10000. So √0.0529 = 23 ÷ 100 = 0.23.
Step 3. Add: 2.3 + 0.23 = 2.53.
Check. 2.3 × 2.3 = 5.29, and 0.23 × 0.23 = 0.0529.
Why this works. The root of a quotient is the quotient of the roots. Dividing by 100 divides the root by 10.
How do you avoid careless mistakes?
What habits keep answers right?
Check the unit digit first. It can remove two or three options at once.
Then square or cube your answer and compare. That one line catches most slips.
Check 5
The square root of 0.0144 is 0.012.
Check 6
The unit digit of the cube root of 1728 is 2.
Exam traps
Decimal trap. A root of a decimal has half the places. √0.0144 is 0.12, not 0.012.
Unit-digit trap. Two digits can give the same square end. Check both before you choose.
Pair trap. For squares, pair the primes. For cubes, group them in threes.
Multiply or divide trap. Read the question twice. They need opposite fixes.
Between trap. Perfect squares between 80 and 280 start at 81, not 64.
MCQ on Square Roots and Cube Roots
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