Maharashtra Board Class 8 Maths Chapter 3 Indices and Cube Root Set 3.2 Solutions

Question 1. Complete the following table.

Answer:

In simple words: The table describes how to express numbers in the form \(a^{\frac{m}{n}}\) as both "power of the root" and "root of the power." For \(a^{\frac{m}{n}}\), it means the \(m^{th}\) power of the \(n^{th}\) root of \(a\), or the \(n^{th}\) root of the \(m^{th}\) power of \(a\).

🎯 Exam Tip: Understanding the relationship between rational indices and roots/powers is crucial for solving problems involving exponents. Pay attention to the numerator (power) and denominator (root) of the fractional index.

Question 2. Write the following numbers in the form of rational indices.
(i) Square root of 5th power of 121.
(ii) Cube of 4th root of 324.
(iii) 5th root of square of 264.
(iv) Cube of cube root of 3.
Answer:
(i) \( (121)^{\frac{5}{2}} \)
(ii) \( (324)^{\frac{3}{4}} \)
(iii) \( (264)^{\frac{2}{5}} \)
(iv) \( (3)^{\frac{3}{3}} \)
In simple words: To convert expressions involving roots and powers into rational indices, the power becomes the numerator and the root becomes the denominator of the fractional exponent. For example, "the \(m^{th}\) power of the \(n^{th}\) root of \(x\)" is written as \(x^{\frac{m}{n}}\).

🎯 Exam Tip: Remember that a square root corresponds to a \(\frac{1}{2}\) power, a cube root to a \(\frac{1}{3}\) power, and so on. The "power" mentioned comes from the numerator of the fraction, and the "root" comes from the denominator.