DAV Class 8 Maths Chapter 2 Brain Teasers
Cubes and Cube Roots Brain Teasers
1. A. Tick (✓) the correct option.
a.
Cube of \(0.1\) is equal to
\(
\begin{aligned}
(i)\,& 1.11 \\[5pt]
(ii)\,& 0.001 \\[5pt]
(iii)\,& 0.101 \\[5pt]
(iv)\,& 0.01
\end{aligned}
\)
Solution
\[ (0.1)^3 = 0.001 \]
Answer \( \color{orange}(ii)\ \color{red} 0.001 \)
b.
The smallest number by which 1944 should be multiplied so that it becomes a perfect cube is
\(
\begin{aligned}
(i)\,& 3 \\[5pt]
(ii)\,& 2 \\[5pt]
(iii)\,& 5 \\[5pt]
(iv)\,& 4
\end{aligned}
\)
Solution
\[ \begin{array}{c|c} 2 & 1944 \\ \hline 2 & 972 \\ \hline 2 & 486 \\ \hline 3 & 243 \\ \hline 3 & 81 \\ \hline 3 & 27 \\ \hline 3 & 9 \\ \hline 3 & 3 \\ \hline & 1 \\ \end{array} \] \[ \begin{aligned} 1944 &= 2^3 \times 3^3 \times 3^2 \\ \text{Required number } &= 3 \\ \end{aligned} \]
Answer \( \color{orange}(i)\ \color{red} 3 \)
c.
Value of \(\sqrt[3]{1000000}\) is
\(
\begin{aligned}
(i)\,& 10 \\[5pt]
(ii)\,& 1000 \\[5pt]
(iii)\,& 100 \\[5pt]
(iv)\,& \text{none of these}
\end{aligned}
\)
Solution
\[ \begin{aligned} & = \sqrt[3]{1000000} \\[5pt] & = \sqrt[3]{10^3 \times 10^3} \\[5pt] &= 10 \times 10 \\[5pt] &= 100 \end{aligned} \]
Answer \( \color{orange}(iii)\ \color{red} 100 \)
d.
\( \sqrt[3]{0.027} - \sqrt[3]{0.008}\) is equal to
\(
\begin{aligned}
(i)\,& 1 \\[5pt]
(ii)\,& 0.1 \\[5pt]
(iii)\,& 0.11 \\[5pt]
(iv)\,& 0.09
\end{aligned}
\)
Solution
\[ \begin{aligned} & = \sqrt[3]{0.027} - \sqrt[3]{0.008} \\[5pt] & = \sqrt[3]{\frac{27}{1000}} - \sqrt[3]{\frac{8}{1000}} \\[5pt] & = \sqrt[3]{\frac{3^3}{10^3}} - \sqrt[3]{\frac{2^3}{10^3}} \\[5pt] & = \frac{3}{10} - \frac{2}{10} \\[5pt] & = \frac{1}{10} \\[5pt] &= 0.1 \end{aligned} \]
Answer\( \color{orange}(ii)\ \color{red} 0.1 \)
e.
Cube of \(\left(\dfrac{-1}{3}\right)\) is equal to
\(
\begin{aligned}
(i)\,& \dfrac{1}{27} \\[5pt]
(ii)\,& \dfrac{-1}{9} \\[5pt]
(iii)\,& \dfrac{-1}{27} \\[5pt]
(iv)\,& \dfrac{1}{9}
\end{aligned}
\)
Solution
\[ \left(\dfrac{-1}{3}\right)^3 = \dfrac{-1}{27} \]
Answer \( \color{orange}(iii)\ \color{red} \dfrac{-1}{27} \)
B. Answer the following questions.
a. Find the number whose cube is 1728
Solution
\[ \begin{array}{c|c} 2 & 1728 \\ \hline 2 & 864 \\ \hline 2 & 432 \\ \hline 2 & 216 \\ \hline 2 & 108 \\ \hline 2 & 54 \\ \hline 3 & 27 \\ \hline 3 & 9 \\ \hline 3 & 3 \\ \hline & 1 \end{array} \]
\[ \begin{aligned} \sqrt[3]{1728} &= \sqrt[3]{2^3 \times 2^3 \times 3^3} \\[5pt] &= 2 \times 2 \times 3\\[5pt] &= 4 \times 3 \\[5pt] &= 12 \end{aligned} \]
Answer \( \color{red}{12} \)
b. Find the value of \(\sqrt[3]{216\times(-125)}\)
Solution
\[ \begin{aligned} &= \sqrt[3]{216 \times (-125)} \\[5pt] &= \sqrt[3]{6^3 \times (-5)^3} \\[5pt] &= 6 \times (-5) \\[5pt] &= -30 \end{aligned} \]
Answer \( \color{red}{-30} \)
c. Find the cube root of \(0.000001\)
Solution
\[ \begin{aligned} & = \sqrt[3]{0.000001} \\[5pt] & = \sqrt[3]{\frac{1}{1000000}} \\[5pt] & = \sqrt[3]{\frac{1}{10^3 \times 10^3}} \\[5pt] & = \frac{1}{10 \times 10} \\[5pt] & = \frac{1}{100} \\[5pt] &= 0.01 \end{aligned} \]
Answer \( \color{red}{0.01} \)
d. What is the smallest number by which 1715 should be divided so that the quotient is a perfect cube?
Solution
\[ \begin{array}{c|c} 5 & 1715 \\ \hline 7 & 343 \\ \hline 7 & 49 \\ \hline 7 & 7 \\ \hline & 1 \end{array} \]
\[ \begin{aligned} 1715 &= 5\times7^3 \\[5pt] \text{Divide by } &= 5 \\ \\ 1715 \div 5 &= 343 \\[5pt] 343 &= 7^3 \end{aligned} \]
Answer Smallest number \( = \color{red}{5} \)
e. Evaluate \(\sqrt[3]{\dfrac{0.512}{0.343}}\)
Solution
\[ \begin{aligned} &= \sqrt[3]{\frac{0.512}{0.343}} \\[5pt] &= \sqrt[3]{\frac{0.512 \times 1000}{0.343 \times 1000}} \\[5pt] &= \sqrt[3]{\frac{512}{343}} \\[5pt] &= \sqrt[3]{\frac{8^3}{7^3}} \\[5pt] &= \frac{8}{7} \\[5pt] \end{aligned} \]
Answer \( \color{red} {\dfrac{8}{7}} \)
2. Prove that if a number is tripled, then its cube is 27 times the cube of the given number.
Solution
\[ \begin{aligned} \text{Let the number be } & = x \\ \text{Triple the number } & = 3x \\ \text{Cube of the number} & = (3x)^3 \\ \\ \implies (3x)^3 & = \ 27 \times x^3 \\ \\ \end{aligned} \]
3. Write cubes of all natural numbers from 1 to 10 and observe the pattern.
Solution
\[ \begin{aligned} 1^3 & = 1 \\ 2^3 & = 8 \implies 3 + 5 \\ 3^3 & = 27 \implies 7 + 9 + 11 \\ 4^3 & = 64 \implies 13 + 15 + 17 + 19 \\ 5^3 & = 125 \implies 21 + 23 + 25 + 27 + 29 \\ 6^3 & = 216 \implies 31 + 33 + 35 + 37 + 39 + 41 \\ 7^3 & = 343 \implies 43 + 45 + 47 + 49 + 51 + 53 + 55 \\ 8^3 & = 512 \implies 57 + 59 + 61 + 63 + 65 + 67 + 69 + 71 \\ 9^3 & = 729 \implies 73 + 75 + 77 + 79 + 81 + 83 + 85 + 87 + 89 \\ 10^3 & = 1000 \implies 91 + 93 + 95 + 97 + 99 + 101 + 103 + 105 + 107 + 109 \\ \end{aligned} \] Cube of a number \( n \) can be expressed as the sum of \( n \) odd consecutive numbers.
4. Find the cubes of
(i) \(0.6^3\)
Solution
\[ \begin{aligned} &= (0.6)^3 \\[5pt] &= 0.6 \times 0.6 \times 0.6 \\[5pt] &= 0.216 \end{aligned} \]
Answer \( \color{red} 0.216\)
(ii) \((-3.1)^3\)
Solution
\[ \begin{aligned} &= (-3.1)^3 \\[5pt] &= (-3.1) \times (-3.1) \times (-3.1) \\[5pt] &= -29.791 \end{aligned} \]
Answer \( \color{red} -29.791\)
(iii) \((-0.01)^3\)
Solution
\[ \begin{aligned} &= (-0.01)^3 \\[5pt] &= (-0.01) \times (-0.01) \times (-0.01) \\[5pt] &= -0.000001 \end{aligned} \]
Answer \(\color{red} -0.000001\)
5. Find the value of the following cube roots:
(i) \(\sqrt[3]{0.008}\)
Solution
\[ \begin{aligned} &= \sqrt[3]{0.008} \\[5pt] &= \sqrt[3]{\frac{8}{1000}} \\[5pt] &= \sqrt[3]{\frac{2^3}{10^3}} \\[5pt] &= \frac{2}{10} \\[5pt] &= 0.2 \end{aligned} \]
Answer \( \color{red}0.2 \)
(ii) \(\sqrt[3]{\dfrac{-64}{1331}}\)
Solution
\[ \begin{aligned} &= \sqrt[3]{\frac{-64}{1331}} \\[5pt] &= \sqrt[3]{\frac{(-4)^3}{11^3}} \\[5pt] &= \frac{-4}{11} \end{aligned} \]
Answer \( \color{red}\dfrac{-4}{11} \)
(iii) \( \sqrt[3]{27} \times \sqrt[3]{2744} \)
Solution
\[ \begin{aligned} \sqrt[3]{27} \times \sqrt[3]{2744} \end{aligned} \] \[ \begin{array}{cc} \begin{array}{c|c} 3 & 27 \\ \hline 3 & 9 \\ \hline 3 & 3 \\ \hline & 1 \\ \end{array} & \quad \begin{array}{c|c} 2 & 2744 \\ \hline 2 & 1372 \\ \hline 2 & 686 \\ \hline 7 & 343 \\ \hline 7 & 49 \\ \hline 7 & 7 \\ \hline & 1 \\ \end{array} \end{array} \] \[ \begin{aligned} & = \sqrt[3]{3^3} \times \sqrt[3]{2^3 \times 7^3} \\ & = 3 \times 2 \times 7 \\ & = 6 \times 7 \\ & = 42 \\ \end{aligned} \]
Answer \( \color{red}42\)
6. Find the smallest number which, when multiplied by 3600, makes the product a perfect cube. Also find the cube root of the product.
Solution
\[ \begin{array}{c|c} 2 & 3600 \\ \hline 2 & 1800 \\ \hline 2 & 900 \\ \hline 2 & 450 \\ \hline 3 & 225 \\ \hline 3 & 75 \\ \hline 5 & 25 \\ \hline 5 & 5 \\ \hline & 1 \end{array} \] \[ \begin{aligned} 3600 &= 2^3 \times 2 \times 3^2 \times 5^2 \\ \\ \text{Required no.} &= 2^2 \times 3 \times 5 \\ &= 4 \times 15 \\ &= 60 \\ \\ \text{Multiply by } &= 60 \\ \\ 3600 \times 60 & = 216000 \\ \\ \sqrt[3]{216000} & = \sqrt[3]{2^3 \times 2^3 \times 3^3 \times 5^3} \\ & = 2 \times 2 \times 3 \times 5 \\ & = 4 \times 15 \\ \sqrt[3]{216000} & = 60 \\ \end{aligned} \]
Answer Smallest number \( = \color{red} 60\), Perfect cube \( = \color{red} 216000\) ,\(\sqrt[3]{216000} = \color{red} 60 \)
7. Evaluate: \( \sqrt[3]{\dfrac{0.027}{0.008}} \div \sqrt{\dfrac{0.09}{0.04}} -1 \)
Solution
\[ \begin{aligned} &= \sqrt[3]{\frac{0.027}{0.008}} \div \sqrt{\frac{0.09}{0.04}} -1 \\ \\ & = \sqrt[3]{\frac{0.027 \times 1000}{0.008 \times 1000}} \div \sqrt{\frac{0.09 \times 100}{0.04 \times 100 }} -1 \\ \\ & = \sqrt[3]{\frac{27}{8}} \div \sqrt{\frac{9}{4}} -1 \\ \\ & = \sqrt[3]{\frac{3^3}{2^3}} \div \sqrt{\frac{3^2}{2^2}} -1 \\ \\ & = \frac{3}{2} \div \frac{3}{2} -1 \\ \\ & = \frac{\cancel3^1}{\cancel2_1} \times \frac{\cancel2^1}{\cancel3_1} -1 \\ \\ & = 1 - 1 \\ & = 0 \end{aligned} \]
Answer \( \color{red} 0\)
8. Guess the cube root of the following numbers:
(i) \( \sqrt[3]{6859} \)
Solution
\[ \begin{aligned} \text{II} \quad & \quad \text{I} \\ 6 \quad & \ \ 859 \end{aligned} \] \[ \begin{aligned} \text{In the first group, }859 &\text{ has 9 in unit place}\\ &\;9^3 = 729\\ \therefore\quad\text{One's place} &= 9 \\[8pt] \text{From the second group : } & 6\\ 1^3 < 6 &< 2^3\\ \therefore\quad\text{Ten's place} &= 1 \\[8pt] \sqrt[3]{6859} & = 19 \end{aligned} \]
Answer \( \sqrt[3]{6859} = \color{red}19 \)
(ii) \( \sqrt[3]{12167} \)
Solution
\[ \begin{aligned} \text{II} \quad & \quad \text{I} \\ 12 \quad & \ \ 167 \end{aligned} \] \[ \begin{aligned} \text{In the first group, }167 &\text{ has 7 in unit place}\\ &\;3^3 = 27\\ \therefore\quad\text{One's place} &= 3 \\[8pt] \text{From the second group: }12\\ 2^3 < 12 &< 3^3\\ \therefore\quad\text{Ten's place} &= 2 \\[8pt] \sqrt[3]{12167} &= 23 \end{aligned} \]
Answer \( \sqrt[3]{12167} = \color{red}23 \)
(iii) \( \sqrt[3]{32768} \)
Solution
\[ \begin{aligned} \text{II} \quad & \quad \text{I} \\ 32 \quad & \ \ 768 \end{aligned} \] \[ \begin{aligned} \text{In the first group, }768 &\text{ has 8 in unit place}\\ \;2^3 &= 8\\ \therefore\quad\text{One's place} &= 2 \\[8pt] \text{From the second group: } & 32\\ 3^3 < 32 &< 4^3\\ \therefore\quad\text{Ten's place} &= 3 \\[8pt] \sqrt[3]{32768} &= 32 \end{aligned} \]
Answer \( \sqrt[3]{32768} = \color{red}32 \)