DAV Class 7 Maths Chapter 4 Worksheet 1
Exponents and Powers Worksheet 1
1. Write the base and exponent in each of the following.
(i) \( \displaystyle \left(-\frac{1}{3}\right)^3\) \( \implies \displaystyle \text{base} = {\color{red} -\frac{1}{3}} , \ \text{exponent} = {\color{red} 3} \)
(ii) \( \displaystyle \left(-\frac{4}{7}\right)^6\) \( \implies \displaystyle \text{base} = {\color{red} -\frac{4}{7}} , \ \text{exponent} = {\color{red} 6} \)
(iii) \( \displaystyle \left(\frac{2}{9}\right)^5\) \( \implies \displaystyle \text{base} = {\color{red} \frac{2}{9}} , \ \text{exponent} = {\color{red} 5} \)
(iv) \( \displaystyle \left(\frac{15}{19}\right)^8\) \( \implies \displaystyle \text{base} = {\color{red} \frac{15}{19}} , \ \text{exponent} = {\color{red} 8} \)
(v) \( (-15)^4 \) \( \implies \displaystyle \text{base} = {\color{red} -15} , \ \text{exponent} = {\color{red} 4} \)
(vi) \( \displaystyle -\frac{2}{3} \) \( \implies \displaystyle \text{base} = {\color{red} -\frac{2}{3}} , \ \text{exponent} = {\color{red} 1} \)
2. Express the following in exponential form.
(i) \( \displaystyle \frac{5}{6} \times \frac{5}{6} \) \( = \displaystyle {\color{red} \left(\frac{5}{6}\right)^2} \)
(ii) \( \displaystyle \frac{9}{2} \times \frac{9}{2} \times \frac{9}{2} \times \frac{9}{2} \) \( = \displaystyle {\color{red} \left(\frac{9}{2}\right)^4} \)
(iii) \( \displaystyle \left(-\frac{7}{8}\right) \times \left(-\frac{7}{8}\right) \times \left(-\frac{7}{8}\right) \) \( = \displaystyle {\color{red} \left(-\frac{7}{8}\right)^3} \)
(iv) \( \displaystyle \left(-\frac{1}{2}\right) \times \left(-\frac{1}{2}\right) \times \left(-\frac{1}{2}\right) \times \left(-\frac{1}{2}\right) \times \left(-\frac{1}{2}\right) \) \( = \displaystyle {\color{red} \left(-\frac{1}{2}\right)^5} \)
(v) \( 1.8 \times 1.8 \times 1.8 \times 1.8 \times 1.8 \times 1.8 \times 1.8 \) \( = \displaystyle {\color{red} (1.8)^7} \)
(vi) \( \displaystyle \frac{11}{12} \times \frac{11}{12} \times \frac{11}{12} \times \frac{11}{12} \times \frac{11}{12} \times \frac{11}{12} \) \( = \displaystyle {\color{red} \left(\frac{11}{12}\right)^6} \)
3. Express the following as rational numbers in the form \( \displaystyle \frac{p}{q} \)
(i) \( \displaystyle {\frac{5}{6}}^3 \)
\begin{align*} & = \frac{5}{6} \times \frac{5}{6} \times \frac{5}{6} \\ \\ & = \color{red} \frac{125}{216} \end{align*}
(ii) \( \displaystyle \left(-\frac{12}{13}\right)^2 \)
\begin{align*} & = \left(-\frac{12}{13}\right) \times \left(-\frac{12}{13}\right) \\ \\ & = \color{red} \frac{144}{169} \end{align*}
(iii) \( \displaystyle \left(\frac{4}{9}\right)^4 \)
\begin{align*} & = \frac{4}{9} \times \frac{4}{9} \times \frac{4}{9} \times \frac{4}{9} \\ \\ & = \color{red} \frac{256}{6561} \end{align*}
(iv) \( \displaystyle \left(-\frac{1}{2}\right)^5 \)
\begin{align*} & = \left(-\frac{1}{2}\right) \times \left(-\frac{1}{2}\right) \times \left(-\frac{1}{2}\right) \times \left(-\frac{1}{2}\right) \times \left(-\frac{1}{2}\right) \\ \\ & = \color{red} \frac{-1}{32} \end{align*}
(v) \( \displaystyle \left(\frac{1}{4}\right)^4 \)
\begin{align*} & = \frac{1}{4} \times \frac{1}{4} \times \frac{1}{4} \times \frac{1}{4} \\ \\ & = \color{red} \frac{1}{256} \end{align*}
(vi) \( \displaystyle \left(\frac{3}{5}\right)^3 \)
\begin{align*} & = \frac{3}{5} \times \frac{3}{5} \times \frac{3}{5} \\ \\ & = \color{red} \frac{27}{125} \end{align*}
4. Express the following as powers of rational numbers.
(i) \( \displaystyle \frac{81}{625} \)
\begin{align*} \text{Factorisation of } 81 \text{ and } 625 \\ \end{align*} \begin{align*} \begin{array}{r|l} 3 & 81 \\ \hline 3 & 27 \\ \hline 3 & 9 \\ \hline 3 & 3 \\ \hline & 1 \\ \end{array} && \begin{array}{r|l} 5 & 625 \\ \hline 5 & 125 \\ \hline 5 & 25 \\ \hline 5 & 5 \\ \hline & 1 \\ \end{array} \end{align*} \begin{align*} \frac{81}{625} &= \frac{3 \times 3 \times 3 \times 3}{5 \times 5 \times 5 \times 5} \\ \\ &= \frac{3^4}{5^4} \\ \\ &= \left(\frac{3}{5}\right)^4 \end{align*}
Answer \( \displaystyle \frac{81}{625} = \color{red} \left(\frac{3}{5}\right)^4 \)
(ii) \( \displaystyle -\frac{8}{125} \)
\begin{align*} \text{Factorisation of } 8 \text{ and } 125 \\ \end{align*} \begin{align*} \begin{array}{r|l} 2 & 8 \\ \hline 2 & 4 \\ \hline 2 & 2 \\ \hline & 1 \\ \end{array} && \begin{array}{r|l} 5 & 125 \\ \hline 5 & 25 \\ \hline 5 & 5 \\ \hline & 1 \\ \end{array} \end{align*} \begin{align*} -\frac{8}{125} &= -\frac{2 \times 2 \times 2}{5 \times 5 \times 5} \\ \\ &= -\frac{2^3}{5^3} \\ \\ &= \left(-\frac{2}{5}\right)^3 \end{align*}
Answer \( \displaystyle -\frac{8}{125} = \color{red} \left(-\frac{2}{5}\right)^3 \)
(iii) \( \displaystyle -\frac{343}{512} \)
\begin{align*} \text{Factorisation of } -343 \text{ and } 512 \\ \end{align*} \begin{align*} \begin{array}{r|l} 7 & 343 \\ \hline 7 & 49 \\ \hline 7 & 7 \\ \hline & -1 \\ \end{array} && \begin{array}{r|l} 2 & 512 \\ \hline 2 & 256 \\ \hline 2 & 128 \\ \hline 2 & 64 \\ \hline 2 & 32 \\ \hline 2 & 16 \\ \hline 2 & 8 \\ \hline 2 & 4 \\ \hline 2 & 2 \\ \hline & 1 \\ \end{array} \end{align*} \begin{align*} \left(-\frac{343}{512}\right) &= -\frac{7 \times 7 \times 7}{\boxed{2 \times 2 \times 2} \times \boxed{2 \times 2 \times 2} \times \boxed{2 \times 2 \times 2}} \\ \\ &= -\frac{7^3}{8 \times 8 \times 8} \\ \\ &= -\frac{7^3}{8^3} \\ \\ &= \left(-\frac{7}{8}\right)^3 \end{align*}
Answer \( \displaystyle -\frac{343}{512} = \color{red} \left(-\frac{7}{8}\right)^3 \)
(iv) \( \displaystyle \frac{32}{243} \)
\begin{align*} \text{Prime factorisation of } 32 \text{ and } 243 \\ \end{align*} \begin{align*} \begin{array}{r|l} 2 & 32 \\ \hline 2 & 16 \\ \hline 2 & 8 \\ \hline 2 & 4 \\ \hline 2 & 2 \\ \hline & 1 \\ \end{array} && \begin{array}{r|l} 3 & 243 \\ \hline 3 & 81 \\ \hline 3 & 27 \\ \hline 3 & 9 \\ \hline 3 & 3 \\ \hline & 1 \\ \end{array} \end{align*} \begin{align*} \frac{32}{243} &= \frac{2 \times 2 \times 2 \times 2 \times 2}{3 \times 3 \times 3 \times 3 \times 3} \\ \\ &= \frac{2^5}{3^5} \\ \\ &= \left(\frac{2}{3}\right)^5 \end{align*}
Answer \( \displaystyle \frac{32}{243} = \color{red} \left(\frac{2}{3}\right)^5 \)
(v) \( \displaystyle -\frac{1}{216} \)
\begin{align*} \text{Prime factorisation of } 216 \\ \end{align*} \begin{align*} \begin{array}{r|l} 2 & 216 \\ \hline 2 & 108 \\ \hline 2 & 54 \\ \hline 3 & 27 \\ \hline 3 & 9 \\ \hline 3 & 3 \\ \hline & 1 \\ \end{array} \end{align*} \begin{align*} -\frac{1}{216} &= -\frac{1^3}{2 \times 2 \times 2 \times 3 \times 3 \times 3} \\ \\ &= -\frac{1^3}{2^3 \times 3^3} \\ \\ &= -\frac{1^3}{{(2 \times 3)}^3} \\ \\ &= -\frac{1^3}{6^3} \\ \\ &= -\frac{1^3}{6^3} \\ \\ &= \left(-\frac{1}{6}\right)^3 \end{align*}
Answer \( \displaystyle -\frac{1}{216} = \color{red} \left(-\frac{1}{6}\right)^3 \)
(vi) \( \displaystyle \frac{729}{1000} \)
\begin{align*} \text{Prime factorisation of } 729 \text{ and } 1000 \\ \end{align*} \begin{align*} \begin{array}{r|l} 3 & 729 \\ \hline 3 & 243 \\ \hline 3 & 81 \\ \hline 3 & 27 \\ \hline 3 & 9 \\ \hline 3 & 3 \\ \hline & 1 \\ \end{array} && \begin{array}{r|l} 2 & 1000 \\ \hline 2 & 500 \\ \hline 2 & 250 \\ \hline 2 & 125 \\ \hline 5 & 125 \\ \hline 5 & 25 \\ \hline 5 & 5 \\ \hline & 1 \\ \end{array} \end{align*} \begin{align*} \frac{729}{1000} &= \frac{3 \times 3 \times 3 \times 3 \times 3 \times 3}{2 \times 2 \times 2 \times 5 \times 5 \times 5} \\ \\ &= \frac{\boxed{3 \times 3} \times \boxed{3 \times 3} \times \boxed{3 \times 3}}{\boxed{2 \times 5} \times \boxed{2 \times 5} \times \boxed{2 \times 5}} \\ \\ &= \frac{9 \times 9 \times 9}{10 \times 10 \times 10} \\ \\ &= \frac{9^3}{10^3} \\ \\ &= \left(\frac{9}{10}\right)^3 \end{align*}
Answer \( \displaystyle \frac{729}{1000} = \color{red} \left(\frac{9}{10}\right)^3 \)