Maharashtra Board Class 8 Maths Chapter 3 Indices and Cube Root Set 3.1 Solutions

Question 1. Express the following numbers in index form.
(i) Fifth root of 13
(ii) Sixth root of 9
(iii) Square root of 256
(iv) Cube root of 17
(v) Eighth root of 100
(vi) Seventh root of 30
Answer:
(i) \( (13)^{\frac{1}{5}} \)
(ii) \( (9)^{\frac{1}{6}} \)
(iii) \( (256)^{\frac{1}{2}} \)
(iv) \( (17)^{\frac{1}{3}} \)
(v) \( (100)^{\frac{1}{8}} \)
(vi) \( (30)^{\frac{1}{7}} \)
In simple words: To express a root of a number in index form, the 'nth' root of 'a' is written as 'a' raised to the power of 1/n.

🎯 Exam Tip: Understanding the relationship between roots and fractional indices is crucial for accurately converting between these forms.

Question 2. Write in the form 'nth root of a' in each of the following numbers.
(i) \( (81)^{\frac{1}{4}} \)
(ii) \( (49)^{\frac{1}{2}} \)
(iii) \( (15)^{\frac{1}{5}} \)
(iv) \( (512)^{\frac{1}{9}} \)
(v) \( (100)^{\frac{1}{19}} \)
(vi) \( (6)^{\frac{1}{7}} \)
Answer:
(i) Fourth root of 81.
(ii) Square root of 49.
(iii) Fifth root of 15.
(iv) Ninth root of 512.
(v) Nineteenth root of 100.
(vi) Seventh root of 6.
In simple words: To convert a number in fractional index form back to 'nth root' form, the denominator of the fraction represents the 'nth' root, and the base number is the value inside the root.

🎯 Exam Tip: Pay close attention to the denominator of the fractional exponent, as it directly indicates the type of root (square, cube, fourth, etc.).

Maharashtra Board Class 8 Maths Chapter 3 Indices And Cube Root Practice Set 3.1 Intext Questions And Activities

Question. 1. Using laws of indices, write proper numbers in the following boxes. (Textbook pg, no. 14)
(i) \( 3^5 \times 3^2 = 3^{()} \)
(ii) \( 3^7 \div 3^9 = 3^{()} \)
(iii) \( (3^4)^5 = 3^{()} \)
(iv) \( 5^{-3} = \frac{1}{5^{()}} \)
(v) \( 5^0 = () \)
(vi) \( 5^1 = () \)
(vii) \( (5 \times 7)^2 = 5^{()} \times 7^{()} \)
(viii) \( (\frac{5}{7})^3 = \frac{()}{()} \)
(ix) \( (\frac{5}{7})^{-3} = (\frac{()}{()})^3 \)
Answer:
(i) \( 3^5 \times 3^2 = 3^7 \)
(ii) \( 3^7 \div 3^9 = 3^{-2} \)
(iii) \( (3^4)^5 = 3^{20} \)
(iv) \( 5^{-3} = \frac{1}{5^3} \)
(v) \( 5^0 = 1 \)
(vi) \( 5^1 = 5 \)
(vii) \( (5 \times 7)^2 = 5^2 \times 7^2 \)
(viii) \( (\frac{5}{7})^3 = \frac{5^3}{7^3} \)
(ix) \( (\frac{5}{7})^{-3} = (\frac{7}{5})^3 \)
In simple words: This exercise tests your understanding of the basic laws of exponents, including multiplication, division, power of a power, negative exponents, zero exponent, and fractional exponents.

🎯 Exam Tip: Memorizing and correctly applying the fundamental laws of indices is essential for solving problems involving powers and roots efficiently.