Cubes and Cube Roots — Exercise 4.2

Find the cube root of the following number by prime factorisation:

12167

Answer

Given number is 12167.

Finding prime factors of 12167 :

$\begin{array}{l|r} 23 & 12167 \\ \hline 23 & 529 \\ \hline 23 & 23 \\ \hline & 1 \\ \end{array}$

Expressing 12167 into prime factors, we have

12167 = 23 × 23 × 23

$\Rightarrow \sqrt[3]{12167}\\[1em] \Rightarrow \sqrt[3]{23 \times 23 \times 23}\\[1em] \Rightarrow 23$

Hence, cube root of 12167 = 23.

Find the cube root of the following number by prime factorisation:

35937

Answer

Given number is 35937.

Finding prime factors of 35937 :

$\begin{array}{l|r} 3 & 35937 \\ \hline 3 & 11979 \\ \hline 3 & 3993 \\ \hline 11 & 1331 \\ \hline 11 & 121 \\ \hline 11 & 11 \\ \hline & 1 \\ \end{array}$

Expressing 35937 into prime factors, we have

35937 = (3 × 3 × 3) × (11 × 11 × 11)

$\Rightarrow \sqrt[3]{35937}\\[1em] \Rightarrow \sqrt[3]{(3 \times 3 \times 3) \times (11 \times 11 \times 11)}\\[1em] \Rightarrow 3 \times 11\\[1em] \Rightarrow 33$

Hence, cube root of 35937 = 33.

Find the cube root of the following number by prime factorisation:

42875

Answer

Given number is 42875.

Finding prime factors of 42875 :

$\begin{array}{l|r} 5 & 42875 \\ \hline 5 & 8575 \\ \hline 5 & 1715 \\ \hline 7 & 343 \\ \hline 7 & 49 \\ \hline 7 & 7 \\ \hline & 1 \\ \end{array}$

Expressing 42875 into prime factors, we have

42875 = (5 × 5 × 5) × (7 × 7 × 7)

$\Rightarrow \sqrt[3]{42875}\\[1em] \Rightarrow \sqrt[3]{(5 \times 5 \times 5) \times (7 \times 7 \times 7)}\\[1em] \Rightarrow 5 \times 7\\[1em] \Rightarrow 35$

Hence, cube root of 42875 = 35.

Find the cube root of the following number by prime factorisation:

21952

Answer

Given number is 21952.

Finding prime factors of 21952 :

$\begin{array}{l|r} 2 & 21952 \\ \hline 2 & 10976 \\ \hline 2 & 5488 \\ \hline 2 & 2744 \\ \hline 2 & 1372 \\ \hline 2 & 686 \\ \hline 7 & 343 \\ \hline 7 & 49 \\ \hline 7 & 7 \\ \hline & 1 \\ \end{array}$

Expressing 21952 into prime factors, we have

21952 = (2 × 2 × 2) × (2 × 2 × 2) × (7 × 7 × 7)

$\Rightarrow \sqrt[3]{21952}\\[1em] \Rightarrow \sqrt[3]{(2 \times 2 \times 2) \times (2 \times 2 \times 2) \times (7 \times 7 \times 7)}\\[1em] \Rightarrow 2 \times 2 \times 7\\[1em] \Rightarrow 28$

Hence, cube root of 21952 = 28.

Find the cube root of the following number by prime factorisation:

373248

Answer

Given number is 373248.

Finding prime factors of 373248 :

$\begin{array}{l|r} 2 & 373248 \\ \hline 2 & 186624 \\ \hline 2 & 93312 \\ \hline 2 & 46656 \\ \hline 2 & 23328 \\ \hline 2 & 11664 \\ \hline 2 & 5832 \\ \hline 2 & 2916 \\ \hline 2 & 1458 \\ \hline 3 & 729 \\ \hline 3 & 243 \\ \hline 3 & 81 \\ \hline 3 & 27 \\ \hline 3 & 9 \\ \hline 3 & 3 \\ \hline & 1 \\ \end{array}$

Expressing 373248 into prime factors, we have

373248 = (2 × 2 × 2) × (2 × 2 × 2) × (2 × 2 × 2) × (3 × 3 × 3) × (3 × 3 × 3)

$\Rightarrow \sqrt[3]{373248}\\[1em] \Rightarrow \sqrt[3]{(2 \times 2 \times 2) \times (2 \times 2 \times 2) \times (2 \times 2 \times 2) \times (3 \times 3 \times 3) \times (3 \times 3 \times 3)}\\[1em] \Rightarrow 2 \times 2 \times 2 \times 3 \times 3\\[1em] \Rightarrow 72$

Hence, cube root of 373248 = 72.

Find the cube root of the following number by prime factorisation:

32768

Answer

Given number is 32768.

Finding prime factors of 32768 :

$\begin{array}{l|r} 2 & 32768 \\ \hline 2 & 16384 \\ \hline 2 & 8192 \\ \hline 2 & 4096 \\ \hline 2 & 2048 \\ \hline 2 & 1024 \\ \hline 2 & 512 \\ \hline 2 & 256 \\ \hline 2 & 128 \\ \hline 2 & 64 \\ \hline 2 & 32 \\ \hline 2 & 16 \\ \hline 2 & 8 \\ \hline 2 & 4 \\ \hline 2 & 2 \\ \hline & 1 \\ \end{array}$

Expressing 32768 into prime factors, we have

32768 = (2 × 2 × 2) × (2 × 2 × 2) × (2 × 2 × 2) × (2 × 2 × 2) × (2 × 2 × 2)

$\Rightarrow \sqrt[3]{32768}\\[1em] \Rightarrow \sqrt[3]{(2 \times 2 \times 2) \times (2 \times 2 \times 2) \times (2 \times 2 \times 2) \times (2 \times 2 \times 2) \times (2 \times 2 \times 2)}\\[1em] \Rightarrow 2 \times 2 \times 2 \times 2 \times 2\\[1em] \Rightarrow 32$

Hence, cube root of 32768 = 32.

Find the cube root of the following number by prime factorisation:

262144

Answer

Given number is 262144.

Finding prime factors of 262144 :

$\begin{array}{l|r} 2 & 262144 \\ \hline 2 & 131072 \\ \hline 2 & 65536 \\ \hline 2 & 32768 \\ \hline 2 & 16384 \\ \hline 2 & 8192 \\ \hline 2 & 4096 \\ \hline 2 & 2048 \\ \hline 2 & 1024 \\ \hline 2 & 512 \\ \hline 2 & 256 \\ \hline 2 & 128 \\ \hline 2 & 64 \\ \hline 2 & 32 \\ \hline 2 & 16 \\ \hline 2 & 8 \\ \hline 2 & 4 \\ \hline 2 & 2 \\ \hline & 1 \\ \end{array}$

Expressing 262144 into prime factors, we have

262144 = (2 × 2 × 2) × (2 × 2 × 2) × (2 × 2 × 2) × (2 × 2 × 2) × (2 × 2 × 2) × (2 × 2 × 2)

$\Rightarrow \sqrt[3]{262144}\\[1em] \Rightarrow \sqrt[3]{(2 \times 2 \times 2) \times (2 \times 2 \times 2) \times (2 \times 2 \times 2) \times (2 \times 2 \times 2) \times (2 \times 2 \times 2) \times (2 \times 2 \times 2)}\\[1em] \Rightarrow 2 \times 2 \times 2 \times 2 \times 2 \times 2\\[1em] \Rightarrow 64$

Hence, cube root of 262144 = 64.

Find the cube root of the following number by prime factorisation:

157464

Answer

Given number is 157464.

Finding prime factors of 157464 :

$\begin{array}{l|r} 2 & 157464 \\ \hline 2 & 78732 \\ \hline 2 & 39366 \\ \hline 3 & 19683 \\ \hline 3 & 6561 \\ \hline 3 & 2187 \\ \hline 3 & 729 \\ \hline 3 & 243 \\ \hline 3 & 81 \\ \hline 3 & 27 \\ \hline 3 & 9 \\ \hline 3 & 3 \\ \hline & 1 \\ \end{array}$

Expressing 157464 into prime factors, we have

157464 = (2 × 2 × 2) × (3 × 3 × 3) × (3 × 3 × 3) × (3 × 3 × 3)

$\Rightarrow \sqrt[3]{157464}\\[1em] \Rightarrow \sqrt[3]{(2 \times 2 \times 2) \times (3 \times 3 \times 3) \times (3 \times 3 \times 3) \times (3 \times 3 \times 3)}\\[1em] \Rightarrow 2 \times 3 \times 3 \times 3\\[1em] \Rightarrow 54$

Hence, cube root of 157464 = 54.

Find the cube root of the following number:

-250047

Answer

Given number is -250047.

Finding prime factors of 250047 :

$\begin{array}{l|r} 3 & 250047 \\ \hline 3 & 83349 \\ \hline 3 & 27783 \\ \hline 3 & 9261 \\ \hline 3 & 3087 \\ \hline 3 & 1029 \\ \hline 7 & 343 \\ \hline 7 & 49 \\ \hline 7 & 7 \\ \hline & 1 \\ \end{array}$

Expressing 250047 into prime factors, we have

250047 = (3 × 3 × 3) × (3 × 3 × 3) × (7 × 7 × 7)

$\Rightarrow \sqrt[3]{-250047}\\[1em] \Rightarrow -\sqrt[3]{250047}\\[1em] \Rightarrow -\sqrt[3]{(3 \times 3 \times 3) \times (3 \times 3 \times 3) \times (7 \times 7 \times 7)}\\[1em] \Rightarrow -(3 \times 3 \times 7)\\[1em] \Rightarrow -63$

Hence, $\mathbf{\sqrt[3]{-250047} = -63}$.

Find the cube root of the following number:

$\dfrac{-64}{1331}$

Answer

Solving,

$\Rightarrow \sqrt[3]{\dfrac{-64}{1331}}\\[1em] \Rightarrow \dfrac{\sqrt[3]{-64}}{\sqrt[3]{1331}}\\[1em] \Rightarrow \dfrac{-\sqrt[3]{(2 \times 2 \times 2) \times (2 \times 2 \times 2)}}{\sqrt[3]{11 \times 11 \times 11}}\\[1em] \Rightarrow \dfrac{-(2 \times 2)}{11}\\[1em] \Rightarrow \dfrac{-4}{11}$

Hence, $\mathbf{\sqrt[3]{\dfrac{-64}{1331}} = \dfrac{-4}{11}}$.

Find the cube root of the following number:

$4\dfrac{17}{27}$

Answer

Converting mixed fraction to improper fraction:

$4\dfrac{17}{27} = \dfrac{125}{27}$

$\Rightarrow \sqrt[3]{4\dfrac{17}{27}}\\[1em] \Rightarrow \sqrt[3]{\dfrac{125}{27}}\\[1em] \Rightarrow \dfrac{\sqrt[3]{125}}{\sqrt[3]{27}}\\[1em] \Rightarrow \dfrac{\sqrt[3]{5 \times 5 \times 5}}{\sqrt[3]{3 \times 3 \times 3}}\\[1em] \Rightarrow \dfrac{5}{3}\\[1em] \Rightarrow 1\dfrac{2}{3}$

Hence, $\mathbf{\sqrt[3]{4\dfrac{17}{27}} = 1\dfrac{2}{3}}$.

Find the cube root of the following number:

$5\dfrac{1182}{2197}$

Answer

Converting mixed fraction to improper fraction:

$5\dfrac{1182}{2197} = \dfrac{12167}{2197}$

$\Rightarrow \sqrt[3]{5\dfrac{1182}{2197}}\\[1em] \Rightarrow \sqrt[3]{\dfrac{12167}{2197}}\\[1em] \Rightarrow \dfrac{\sqrt[3]{12167}}{\sqrt[3]{2197}}\\[1em] \Rightarrow \dfrac{\sqrt[3]{23 \times 23 \times 23}}{\sqrt[3]{13 \times 13 \times 13}}\\[1em] \Rightarrow \dfrac{23}{13}\\[1em] \Rightarrow 1\dfrac{10}{13}$

Hence, $\mathbf{\sqrt[3]{5\dfrac{1182}{2197}} = 1\dfrac{10}{13}}$.

Evaluate the following:

$\sqrt[3]{512 \times 729}$

Answer

Evaluating,

$\Rightarrow \sqrt[3]{512 \times 729}\\[1em] \Rightarrow \sqrt[3]{512} \times \sqrt[3]{729}\\[1em] \Rightarrow \sqrt[3]{(2 \times 2 \times 2) \times (2 \times 2 \times 2) \times (2 \times 2 \times 2)} \times \sqrt[3]{(3 \times 3 \times 3) \times (3 \times 3 \times 3)}\\[1em] \Rightarrow (2 \times 2 \times 2) \times (3 \times 3)\\[1em] \Rightarrow 8 \times 9\\[1em] \Rightarrow 72$

Hence, $\mathbf{\sqrt[3]{512 \times 729} = 72}$.

Evaluate the following:

$\sqrt[3]{(-1331) \times (3375)}$

Answer

Evaluating,

$\Rightarrow \sqrt[3]{(-1331) \times (3375)}\\[1em] \Rightarrow \sqrt[3]{-1331} \times \sqrt[3]{3375}\\[1em] \Rightarrow -\sqrt[3]{11 \times 11 \times 11} \times \sqrt[3]{(3 \times 3 \times 3) \times (5 \times 5 \times 5)}\\[1em] \Rightarrow (-11) \times (3 \times 5)\\[1em] \Rightarrow (-11) \times 15\\[1em] \Rightarrow -165$

Hence, $\mathbf{\sqrt[3]{(-1331) \times (3375)} = -165}$.

Find the cube root of the following decimal number:

0.003375

Answer

Converting decimal to fraction:

$0.003375 = \dfrac{3375}{1000000}$

$\Rightarrow \sqrt[3]{0.003375}\\[1em] \Rightarrow \sqrt[3]{\dfrac{3375}{1000000}}\\[1em] \Rightarrow \dfrac{\sqrt[3]{3375}}{\sqrt[3]{1000000}}\\[1em] \Rightarrow \dfrac{\sqrt[3]{(3 \times 3 \times 3) \times (5 \times 5 \times 5)}}{\sqrt[3]{100 \times 100 \times 100}}\\[1em] \Rightarrow \dfrac{3 \times 5}{100}\\[1em] \Rightarrow \dfrac{15}{100}\\[1em] \Rightarrow 0.15$

Hence, $\mathbf{\sqrt[3]{0.003375} = 0.15}$.

Find the cube root of the following decimal number:

19.683

Answer

Converting decimal to fraction:

$19.683 = \dfrac{19683}{1000}$

$\Rightarrow \sqrt[3]{19.683}\\[1em] \Rightarrow \sqrt[3]{\dfrac{19683}{1000}}\\[1em] \Rightarrow \dfrac{\sqrt[3]{19683}}{\sqrt[3]{1000}}\\[1em] \Rightarrow \dfrac{\sqrt[3]{(3 \times 3 \times 3) \times (3 \times 3 \times 3) \times (3 \times 3 \times 3)}}{\sqrt[3]{10 \times 10 \times 10}}\\[1em] \Rightarrow \dfrac{3 \times 3 \times 3}{10}\\[1em] \Rightarrow \dfrac{27}{10}\\[1em] \Rightarrow 2.7$

Hence, $\mathbf{\sqrt[3]{19.683} = 2.7}$.

Evaluate: $\sqrt[3]{27} + \sqrt[3]{0.008} + \sqrt[3]{0.064}$

Answer

Evaluating,

$\Rightarrow \sqrt[3]{27} + \sqrt[3]{0.008} + \sqrt[3]{0.064}\\[1em] \Rightarrow \sqrt[3]{3 \times 3 \times 3} + \sqrt[3]{\dfrac{8}{1000}} + \sqrt[3]{\dfrac{64}{1000}}\\[1em] \Rightarrow 3 + \dfrac{\sqrt[3]{2 \times 2 \times 2}}{\sqrt[3]{10 \times 10 \times 10}} + \dfrac{\sqrt[3]{4 \times 4 \times 4}}{\sqrt[3]{10 \times 10 \times 10}}\\[1em] \Rightarrow 3 + \dfrac{2}{10} + \dfrac{4}{10}\\[1em] \Rightarrow 3 + 0.2 + 0.4\\[1em] \Rightarrow 3.6$

Hence, $\mathbf{\sqrt[3]{27} + \sqrt[3]{0.008} + \sqrt[3]{0.064} = 3.6}$.

Multiply 6561 by the smallest number so that product is a perfect cube. Also find the cube root of the product.

Answer

Finding prime factors of 6561 :

$\begin{array}{l|r} 3 & 6561 \\ \hline 3 & 2187 \\ \hline 3 & 729 \\ \hline 3 & 243 \\ \hline 3 & 81 \\ \hline 3 & 27 \\ \hline 3 & 9 \\ \hline 3 & 3 \\ \hline & 1 \\ \end{array}$

6561 = (3 × 3 × 3) × (3 × 3 × 3) × 3 × 3

Since 3 × 3 is left after grouping the prime factors in triplets, so one more 3 is needed to complete the triplet.

∴ 6561 should be multiplied by 3.

Product = 6561 × 3 = 19683

$\Rightarrow \sqrt[3]{19683}\\[1em] \Rightarrow \sqrt[3]{(3 \times 3 \times 3) \times (3 \times 3 \times 3) \times (3 \times 3 \times 3)}\\[1em] \Rightarrow 3 \times 3 \times 3\\[1em] \Rightarrow 27$

Hence, 6561 should be multiplied by 3 to get the perfect cube 19683 and $\mathbf{\sqrt[3]{19683} = 27}$.

Divide the number 8748 by the smallest number so that the quotient is a perfect cube. Also find the cube root of the quotient.

Answer

Finding prime factors of 8748 :

$\begin{array}{l|r} 2 & 8748 \\ \hline 2 & 4374 \\ \hline 3 & 2187 \\ \hline 3 & 729 \\ \hline 3 & 243 \\ \hline 3 & 81 \\ \hline 3 & 27 \\ \hline 3 & 9 \\ \hline 3 & 3 \\ \hline & 1 \\ \end{array}$

8748 = 2 × 2 × (3 × 3 × 3) × (3 × 3 × 3) × 3

Since 2 × 2 × 3 is left after grouping the prime factors in triplets, so 8748 should be divided by 2 × 2 × 3 = 12.

Quotient = 8748 ÷ 12 = 729

$\Rightarrow \sqrt[3]{729}\\[1em] \Rightarrow \sqrt[3]{(3 \times 3 \times 3) \times (3 \times 3 \times 3)}\\[1em] \Rightarrow 3 \times 3\\[1em] \Rightarrow 9$

Hence, 8748 should be divided by 12 to get the perfect cube 729 and $\mathbf{\sqrt[3]{729} = 9}$.

The volume of a cubical box is 21952 cm³. Find the length of side of the box.

Answer

Let the length of the side of the cubical box be $a$ cm.

We know that, volume of a cube = (side)³

Finding prime factors of 21952 :

$\begin{array}{l|r} 2 & 21952 \\ \hline 2 & 10976 \\ \hline 2 & 5488 \\ \hline 2 & 2744 \\ \hline 2 & 1372 \\ \hline 2 & 686 \\ \hline 7 & 343 \\ \hline 7 & 49 \\ \hline 7 & 7 \\ \hline & 1 \\ \end{array}$

21952 = (2 × 2 × 2) × (2 × 2 × 2) × (7 × 7 × 7)

As, volume of the cube = 21952 cm³

$\Rightarrow a^3 = 21952\\[1em] \Rightarrow a = \sqrt[3]{21952}\\[1em] \Rightarrow a = \sqrt[3]{(2 \times 2 \times 2) \times (2 \times 2 \times 2) \times (7 \times 7 \times 7)}\\[1em] \Rightarrow a = 2 \times 2 \times 7\\[1em] \Rightarrow a = 28$

Hence, the length of side of the box is 28 cm.

Three numbers are in the ratio 3 : 4 : 5. If their product is 480, find the numbers.

Answer

Let the three numbers be $3x, 4x$ and $5x$.

$\Rightarrow 3x \times 4x \times 5x = 480\\[1em] \Rightarrow 60x^3 = 480\\[1em] \Rightarrow x^3 = \dfrac{480}{60}\\[1em] \Rightarrow x^3 = 8\\[1em] \Rightarrow x = \sqrt[3]{8}\\[1em] \Rightarrow x = \sqrt[3]{2 \times 2 \times 2}\\[1em] \Rightarrow x = 2\\[1em] \Rightarrow 3x = 3 \times 2 = 6\\[1em] \Rightarrow 4x = 4 \times 2 = 8\\[1em] \Rightarrow 5x = 5 \times 2 = 10$

Hence, the three numbers are 6, 8 and 10.

Two numbers are in the ratio 4 : 5. If difference of their cubes is 61, find the numbers.

Answer

Let the two numbers be $4x$ and $5x$.

$\Rightarrow (5x)^3 - (4x)^3 = 61\\[1em] \Rightarrow 125x^3 - 64x^3 = 61\\[1em] \Rightarrow 61x^3 = 61\\[1em] \Rightarrow x^3 = \dfrac{61}{61}\\[1em] \Rightarrow x^3 = 1\\[1em] \Rightarrow x = \sqrt[3]{1}\\[1em] \Rightarrow x = 1\\[1em] \Rightarrow 4x = 4 \times 1 = 4\\[1em] \Rightarrow 5x = 5 \times 1 = 5$

Hence, the two numbers are 4 and 5.

Difference of two perfect cubes is 387. If the cube root of the greater of two numbers is 8, find the cube root of the smaller number.

Answer

Cube root of the greater number = 8

∴ Greater number = 8³ = 512

Let the cube root of the smaller number be $x$.

∴ Smaller number = $x^3$

$\Rightarrow 512 - x^3 = 387\\[1em] \Rightarrow x^3 = 512 - 387\\[1em] \Rightarrow x^3 = 125\\[1em] \Rightarrow x = \sqrt[3]{125}\\[1em] \Rightarrow x = \sqrt[3]{5 \times 5 \times 5}\\[1em] \Rightarrow x = 5$

Hence, the cube root of the smaller number is 5.