Maharashtra Board Class 8 Maths Chapter 3 Indices and Cube Root Set 3.3 Solutions

Question 1. Find the cube root of the following numbers.
(i) 8000
(ii) 729
(iii) 343
(iv) -512
(v) -2744
(vi) 32768
Answer:
(i) 8000 \[ \begin{align*} & = 2 \times 2 \times 2 \times 10 \times 10 \times 10 \\ & = (2 \times 10) \times (2 \times 10) \times (2 \times 10) \\ & = (2 \times 10)^3 \\ & = 20^3 \end{align*} \]
\( \implies \sqrt[3]{8000} = 20 \)

(ii) 729 \[ \begin{align*} & = (3 \times 3) \times (3 \times 3) \times (3 \times 3) \\ & = (3 \times 3)^3 \\ & = 9^3 \end{align*} \]
\( \implies \sqrt[3]{729} = 9 \)

(iii) 343 \[ \begin{align*} & = 7 \times 7 \times 7 \\ & = 7^3 \end{align*} \]
\( \implies \sqrt[3]{343} = 7 \)

(iv) -512 \[ \begin{align*} & = 2 \times 2 \times 2 \times 4 \times 4 \times 4 \\ & = (2 \times 4) \times (2 \times 4) \times (2 \times 4) \\ & = (2 \times 4)^3 \\ & = 8^3 \end{align*} \]
\( \implies -512 = (-8) \times (-8) \times (-8) \) \( = (-8)^3 \)
\( \implies \sqrt[3]{-512} = -8 \)

(v) -2744 \[ \begin{align*} & = 2 \times 2 \times 2 \times 7 \times 7 \times 7 \\ & = (2 \times 7) \times (2 \times 7) \times (2 \times 7) \\ & = (2 \times 7)^3 \\ & = 14^3 \end{align*} \]
\( \implies -2744 = (-14) \times (-14) \times (-14) \) \( = (-14)^3 \)
\( \implies \sqrt[3]{-2744} = -14 \)

(vi) 32768 \[ \begin{align*} & = 2 \times 2 \times 2 \times 4 \times 4 \times 4 \times 4 \times 4 \times 4 \\ & = (2 \times 4 \times 4) \times (2 \times 4 \times 4) \times (2 \times 4 \times 4) \\ & = (2 \times 4 \times 4)^3 \\ & = 32^3 \end{align*} \]
\( \implies \sqrt[3]{32768} = 32 \)

In simple words: The cube root of a number is found by determining which number, when multiplied by itself three times, results in the original number. For negative numbers, the cube root is also negative.

🎯 Exam Tip: When finding cube roots, always use prime factorization or factor into cubes to ensure accuracy, especially for larger numbers. Remember that the cube root of a negative number is negative.

Question 2. Simplify:
(i) \(\sqrt[3]{\frac{27}{125}}\)
(ii) \(\sqrt[3]{\frac{16}{54}}\)
(iii) If \(\sqrt[3]{729} = 9\) then \(\sqrt[3]{0.000729} = ?\)
Answer:
(i) \(\sqrt[3]{\frac{27}{125}}\) \[ \begin{align*} & = \frac{\sqrt[3]{27}}{\sqrt[3]{125}} \quad \left[ \because \sqrt[m]{\frac{a}{b}} = \frac{\sqrt[m]{a}}{\sqrt[m]{b}} \right] \\ & = \frac{\sqrt[3]{3 \times 3 \times 3}}{\sqrt[3]{5 \times 5 \times 5}} \\ & = \frac{\sqrt[3]{3^3}}{\sqrt[3]{5^3}} \\ & = \frac{(3^3)^{\frac{1}{3}}}{(5^3)^{\frac{1}{3}}} \end{align*} \]
\( \implies \sqrt[3]{\frac{27}{125}} = \frac{3}{5} \quad \left[ \because (a^m)^{\frac{1}{m}} = a \right] \)
(ii) \(\sqrt[3]{\frac{16}{54}}\) \[ \begin{align*} & = \sqrt[3]{\frac{8 \times 2}{27 \times 2}} \\ & = \sqrt[3]{\frac{8}{27}} \\ & = \frac{\sqrt[3]{8}}{\sqrt[3]{27}} \quad \left[ \because \sqrt[m]{\frac{a}{b}} = \frac{\sqrt[m]{a}}{\sqrt[m]{b}} \right] \\ & = \frac{\sqrt[3]{2 \times 2 \times 2}}{\sqrt[3]{3 \times 3 \times 3}} \\ & = \frac{\sqrt[3]{2^3}}{\sqrt[3]{3^3}} \\ & = \frac{(2^3)^{\frac{1}{3}}}{(3^3)^{\frac{1}{3}}} \end{align*} \]
\( \implies \sqrt[3]{\frac{16}{54}} = \frac{2}{3} \quad \left[ \because (a^m)^{\frac{1}{m}} = a \right] \)
(iii) \(\sqrt[3]{0.000729}\) \[ \begin{align*} & = \sqrt[3]{\frac{729}{1000000}} \\ & = \frac{\sqrt[3]{729}}{\sqrt[3]{1000000}} \quad \left[ \because \sqrt[m]{\frac{a}{b}} = \frac{\sqrt[m]{a}}{\sqrt[m]{b}} \right] \\ & = \frac{9}{\sqrt[3]{100^3}} \quad [\because \sqrt[3]{729} = 9] \\ & = \frac{9}{(100^3)^{\frac{1}{3}}} \end{align*} \]
\( \implies \sqrt[3]{0.000729} = \frac{9}{100} \quad \left[ \because (a^m)^{\frac{1}{m}} = a \right] \)
\( \implies \sqrt[3]{0.000729} = 0.09 \)
Note:
Here, number of decimal places in cube root = 6
\( \implies \) number of decimal places in cube of number = 2 In simple words: Simplifying cube root expressions involves factorizing the numbers inside the root and applying the property that the cube root of a fraction is the cube root of the numerator divided by the cube root of the denominator. For decimals, convert to a fraction before finding the cube root.

🎯 Exam Tip: Remember the properties of cube roots for fractions and decimals. When simplifying fractions under a cube root, try to cancel common factors first. For decimal cube roots, convert the decimal to a fraction to simplify the process, paying attention to the number of decimal places.

Maharashtra Board Class 8 Maths Chapter 3 Indices And Cube Root Practice Set 3.3 Intext Questions And Activities

Question 1. 17 is a positive number. The cube of 17, which is 4913, is also a positive number. Cube of -6 is -216. Take some more positive and negative numbers and obtain their cubes. Find the relation between the sign of a number and the sign of its cube. (Textbook pg. no. 17)
Answer:
Solution:
Consider, \(6^3 = 6 \times 6 \times 6 = 216\) and \((-4)^3 = (-4) \times (-4) \times (-4) = -64\)
Thus, cube of a positive number is positive and cube of a negative number is negative.
\( \implies \) Sign of a number = sign of its cube. In simple words: The sign of a cube is the same as the sign of the original number. Positive numbers have positive cubes, and negative numbers have negative cubes.

🎯 Exam Tip: This concept highlights a fundamental property of odd powers. Students should be able to quickly determine the sign of a number raised to an odd power. This is crucial for solving equations and understanding number properties.

Question 2. In example 4 and 5 on textbook pg. no. 17, observe the number of decimal places in the number and number of decimal places in the cube of the number. Is there any relation between the two? (Textbook pg. no. 17)
Answer:
Solution:
Yes, there is a relation between the number of decimal places in the number and its cube.
\((1.2)^3 = 1.728\), \((0.02)^3 = 0.000008\)
No. of decimal places in 1.2 = 1
No. of decimal places in 1.728 = 3
No. of decimal places in 0.02 = 2
No. of decimal places in 0.000008 = 6
Thus, number of decimal places in cube of a number is three times the number of decimal places in that number. In simple words: When you cube a number with decimals, the number of decimal places in the result will be three times the number of decimal places in the original number.

🎯 Exam Tip: This rule is essential for accurately performing calculations with decimals and understanding the magnitude of cubed decimal numbers. Practice with various examples to solidify this understanding.