Sec 1 Maths · Primes, HCF and LCM

Squares, cubes and their roots questions with answers

Practice questions on squares, cubes and their roots, from the basics up to exam level, each with a full worked solution. Try them on paper first, then open the solution to check your working.

Worked example

Express $5184$ as a product of its prime factors. Hence find $\sqrt{5184}$ without using a calculator.

Step 1 of 3

Short division of 5184 by primes, smallest first, until you reach 1

\def\arraystretch{1.8}\begin{array}{r|l} 2 & \underline{\;5184\;} \\ 2 & \underline{\;2592\;} \\ 2 & \underline{\;1296\;} \\ 2 & \underline{\;648\;} \\ 2 & \underline{\;324\;} \\ 2 & \underline{\;162\;} \\ 3 & \underline{\;81\;} \\ 3 & \underline{\;27\;} \\ 3 & \underline{\;9\;} \\ 3 & \underline{\;3\;} \\ & \;1 \end{array}

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Questions

Question 1Square root by prime factorisation
Express $8100$ as a product of its prime factors. Hence find $\sqrt{8100}$ without using a calculator.
Show the worked solution
1. Step 1: Short division of 8100 by primes, smallest first, until you reach 1 $\def\arraystretch{1.8}\begin{array}{r|l} 2 & \underline{\;8100\;} \\ 2 & \underline{\;4050\;} \\ 3 & \underline{\;2025\;} \\ 3 & \underline{\;675\;} \\ 3 & \underline{\;225\;} \\ 3 & \underline{\;75\;} \\ 5 & \underline{\;25\;} \\ 5 & \underline{\;5\;} \\ & \;1 \end{array}$
2. Step 2: Write the result in index notation $8100 = 2^{2} \times 3^{4} \times 5^{2}$
3. Step 3: Every power is even: halve each power to take the square root $\sqrt{8100} = 2 \times 3^{2} \times 5$ $= 90$
Question 2Cube root by prime factorisation
It is given that $13824 = 2^{9} \times 3^{3}$ . Without using a calculator, find $\sqrt[3]{13824}$ .
Show the worked solution
1. Step 1: Every power is a multiple of 3: divide each power by 3 to take the cube root $\sqrt[3]{13824} = 2^{3} \times 3$ $= 24$
Question 3Smallest k for a perfect square
Find the smallest positive whole number $k$ such that $54k$ is a perfect square.
Show the worked solution
1. Step 1: Short division of 54 by primes, smallest first, until you reach 1; then 54 in index notation $\def\arraystretch{1.8}\begin{array}{r|l} 2 & \underline{\;54\;} \\ 3 & \underline{\;27\;} \\ 3 & \underline{\;9\;} \\ 3 & \underline{\;3\;} \\ & \;1 \end{array}$ $54 = 2 \times 3^{3}$
2. Step 2: In a perfect square every power is even: multiply by the primes whose powers are odd $k = 2 \times 3$ $= 6$ Check: $54k = 2^{2} \times 3^{4}$ , and every power is even.
Question 4Smallest k for a perfect cube
Find the smallest positive whole number $k$ such that $375k$ is a perfect cube.
Show the worked solution
1. Step 1: Short division of 375 by primes, smallest first, until you reach 1; then 375 in index notation $\def\arraystretch{1.8}\begin{array}{r|l} 3 & \underline{\;375\;} \\ 5 & \underline{\;125\;} \\ 5 & \underline{\;25\;} \\ 5 & \underline{\;5\;} \\ & \;1 \end{array}$ $375 = 3 \times 5^{3}$
2. Step 2: In a perfect cube every power is a multiple of 3: multiply by enough of each prime to reach the next multiple of 3 $k = 3^{2}$ $= 9$ Check: $375k = 3^{3} \times 5^{3}$ , and every power is a multiple of 3.
Question 5Smallest k to divide by for a square or cube
It is given that $288 = 2^{5} \times 3^{2}$ . Find the smallest positive whole number $k$ such that $\dfrac{288}{k}$ is a perfect square.
Show the worked solution
1. Step 1: Dividing takes powers away: remove just enough of each prime to leave an EVEN power $k = 2$ $= 2$ Check: $\dfrac{288}{k} = 2^{4} \times 3^{2}$ , and every power is even.
Question 6Is it a perfect square or cube? Give the reason
It is given that $2000 = 2^{4} \times 5^{3}$ . Using this, explain whether $2000$ is a perfect cube. Choose the answer with its reason.
Show the worked solution
1. Step 1: A perfect cube has every index a MULTIPLE OF 3 (its cube root divides them by 3)Index of $2$ : $4$ is not a multiple of 3.Index of $5$ : $3$ is a multiple of 3.
2. Step 2: So the answer, with its reasonNo, $2000$ is not a perfect cube: the index of $2$ is not a multiple of 3.

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Worked example

Express

5184

as a product of its prime factors. Hence find

\sqrt{5184}

without using a calculator.

Step 1 of 3

Short division of 5184 by primes, smallest first, until you reach 1

\def\arraystretch{1.8}\begin{array}{r|l} 2 & \underline{\;5184\;} \\ 2 & \underline{\;2592\;} \\ 2 & \underline{\;1296\;} \\ 2 & \underline{\;648\;} \\ 2 & \underline{\;324\;} \\ 2 & \underline{\;162\;} \\ 3 & \underline{\;81\;} \\ 3 & \underline{\;27\;} \\ 3 & \underline{\;9\;} \\ 3 & \underline{\;3\;} \\ & \;1 \end{array}

Questions

Question 1Square root by prime factorisation

Express $8100$ as a product of its prime factors. Hence find $\sqrt{8100}$ without using a calculator.

Show the worked solution

Step 1: Short division of 8100 by primes, smallest first, until you reach 1 $\def\arraystretch{1.8}\begin{array}{r|l} 2 & \underline{\;8100\;} \\ 2 & \underline{\;4050\;} \\ 3 & \underline{\;2025\;} \\ 3 & \underline{\;675\;} \\ 3 & \underline{\;225\;} \\ 3 & \underline{\;75\;} \\ 5 & \underline{\;25\;} \\ 5 & \underline{\;5\;} \\ & \;1 \end{array}$
Step 2: Write the result in index notation $8100 = 2^{2} \times 3^{4} \times 5^{2}$
Step 3: Every power is even: halve each power to take the square root $\sqrt{8100} = 2 \times 3^{2} \times 5$ $= 90$

Question 2Cube root by prime factorisation

It is given that $13824 = 2^{9} \times 3^{3}$ . Without using a calculator, find $\sqrt[3]{13824}$ .

Show the worked solution

Step 1: Every power is a multiple of 3: divide each power by 3 to take the cube root $\sqrt[3]{13824} = 2^{3} \times 3$ $= 24$

Question 3Smallest k for a perfect square

Find the smallest positive whole number $k$ such that $54k$ is a perfect square.

Show the worked solution

Step 1: Short division of 54 by primes, smallest first, until you reach 1; then 54 in index notation $\def\arraystretch{1.8}\begin{array}{r|l} 2 & \underline{\;54\;} \\ 3 & \underline{\;27\;} \\ 3 & \underline{\;9\;} \\ 3 & \underline{\;3\;} \\ & \;1 \end{array}$ $54 = 2 \times 3^{3}$
Step 2: In a perfect square every power is even: multiply by the primes whose powers are odd $k = 2 \times 3$ $= 6$ Check: $54k = 2^{2} \times 3^{4}$ , and every power is even.

Question 4Smallest k for a perfect cube

Find the smallest positive whole number $k$ such that $375k$ is a perfect cube.

Show the worked solution

Step 1: Short division of 375 by primes, smallest first, until you reach 1; then 375 in index notation $\def\arraystretch{1.8}\begin{array}{r|l} 3 & \underline{\;375\;} \\ 5 & \underline{\;125\;} \\ 5 & \underline{\;25\;} \\ 5 & \underline{\;5\;} \\ & \;1 \end{array}$ $375 = 3 \times 5^{3}$
Step 2: In a perfect cube every power is a multiple of 3: multiply by enough of each prime to reach the next multiple of 3 $k = 3^{2}$ $= 9$ Check: $375k = 3^{3} \times 5^{3}$ , and every power is a multiple of 3.

Question 5Smallest k to divide by for a square or cube

It is given that $288 = 2^{5} \times 3^{2}$ . Find the smallest positive whole number $k$ such that $\dfrac{288}{k}$ is a perfect square.

Show the worked solution

Step 1: Dividing takes powers away: remove just enough of each prime to leave an EVEN power $k = 2$ $= 2$ Check: $\dfrac{288}{k} = 2^{4} \times 3^{2}$ , and every power is even.

Question 6Is it a perfect square or cube? Give the reason

It is given that $2000 = 2^{4} \times 5^{3}$ . Using this, explain whether $2000$ is a perfect cube. Choose the answer with its reason.

Show the worked solution

Step 1: A perfect cube has every index a MULTIPLE OF 3 (its cube root divides them by 3)Index of $2$ : $4$ is not a multiple of 3.Index of $5$ : $3$ is a multiple of 3.
Step 2: So the answer, with its reasonNo, $2000$ is not a perfect cube: the index of $2$ is not a multiple of 3.