DAV Class 8 Maths Chapter 3 Worksheet 1
Exponents and Radicals Worksheet 1
1. Express each of the following in exponential form:
(i) \( \sqrt[\large5]{35} \)
Answer \( \color{red} (35)^{\tfrac{1}{5}} \)
(ii) \( \sqrt[11]{ \left(27 \right)^2} \)
Answer \( \color{red} (27)^{\tfrac{2}{11}} \)
(iii) \( \sqrt[\Large 7]{\dfrac{11}{3}} \)
Answer \( \color{red} \left( \dfrac{11}{3} \right)^{\tfrac{1}{7}} \)
(iv) \( \displaystyle \sqrt[\Large 3]{\left( \frac{2}{5} \right)^{-3}} \)
\[ \begin{align*} = \left( \frac{2}{5} \right)^{ \tfrac{-3}{3}} \\ \\ = \left( \frac{2}{5} \right)^{ \frac{-\cancel{3}^1}{\cancel3_1}} \\ \\ = \left( \frac{2}{5} \right)^{-1} \end{align*} \]
Answer \( \color{red} \left( \dfrac{2}{5} \right)^{-1} \)
(v) \( \sqrt[13]{ \left(111 \right)^3} \)
Answer \( \color{red} (111)^{\tfrac{3}{13}} \)
(vi) \( \sqrt[\large7]{\left(29 \right)^2} \)
Answer \( \color{red} (29)^{\tfrac{2}{7}} \)
(vii) \( \sqrt[\large 3]{\left( 2 \right)^{-6}} \)
\[ \begin{align*} = \left( 2 \right)^{ \tfrac{-6}{3}} \\ \\ = \left( 2 \right)^{ \frac{-\cancel{6}^2}{\cancel3_1}} \\ \\ = \left( 2 \right)^{-2} \end{align*} \]
Answer \( \color{red} \left( 2 \right)^{-2} \)
(viii) \( \displaystyle \sqrt[\Large 7]{\left( \frac{15}{341} \right)^{-3}} \)
Answer \( \color{red} \left( \dfrac{15}{341} \right)^{\tfrac{-3}{7}} \)
2. Express each of the following as radicals:
(i) \( (21)^{\tfrac{1}{8}} \)
Answer \( \color{red} \sqrt[\Large 8]{21} \)
(ii) \( (25)^{\tfrac{3}{4}} \)
Answer \( \color{red} \sqrt[\Large 4]{\left(25 \right)^{3}} \)
(iii) \( \left( \dfrac{2}{9} \right)^{\tfrac{1}{9}} \)
Answer \( \color{red} \sqrt[\Large 9]{ \dfrac{2}{9} } \)
(iv) \( (100)^{\tfrac{-5}{6}} \)
\[ \begin{align*} = (100)^{\tfrac{-5}{6}} \\ \\ = \frac{1}{(100)^{\tfrac{5}{6}}} \\ \\ = \sqrt[\Large 6]{\frac{1}{(100)^{5}}} \\ \\ \end{align*} \]
Answer \( \color{red} \sqrt[\Large 6]{\dfrac{1}{(100)^{5}}} \)
(v) \( \left( \dfrac{8}{9} \right)^{\tfrac{3}{4}} \)
Answer \( \color{red} \sqrt[\Large 4]{\left(\dfrac{8}{9}\right)^3} \)
(vi) \( \left( \dfrac{17}{231} \right)^{\tfrac{-5}{6}} \)
\[ \begin{align*} = \left( \frac{17}{231} \right)^{\tfrac{-5}{6}} \\ \\ = \left( \frac{231}{17} \right)^{\tfrac{5}{6}} \\ \\ = \sqrt[\Large 6]{ \left( \frac{231}{17} \right)^{5} } \end{align*} \]
Answer \( \color{red} \sqrt[\Large 6]{ \left( \dfrac{231}{17} \right)^{5} } \)
(vii) \( \left( \dfrac{15}{21} \right)^{\tfrac{2}{5}} \)
Answer \( \color{red} \sqrt[\Large 5]{ \left( \dfrac{15}{21} \right)^2 } \)
3. Express each of the following with positive exponent.
(i) \( x^{\tfrac{-1}{2}} \)
Answer \( \color{red} \dfrac{1}{x^{\tfrac{1}{2}}} \)
(ii) \( x^{\tfrac{-2}{5}} \)
Answer \( \color{red} \dfrac{1}{x^{\tfrac{2}{5}}} \)
(iii) \( \dfrac{7}{x^{\tfrac{-5}{6}}} \)
Answer \( \color{red} 7{x^{\tfrac{5}{6}}} \)
(iv) \( (x^{-3})^4 \)
Answer \( \color{red} \dfrac{1}{(x)^{12}} \)