DAV Class 7 Maths Chapter 4 Worksheet 3

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DAV Class 7 Maths Chapter 4 Worksheet 3

Exponents and Powers Worksheet 3


1. Which of the following statements are true?

(i) \( \displaystyle \left(+\frac{3}{4}\right)^3 \times \left(-\frac{3}{4}\right)^3 = \left(-\frac{3}{4}\right)^6 \)

Answer False

Base is not the same.

(ii) \( \displaystyle \left(\frac{4}{7}\right)^5 \times \left(\frac{4}{7}\right)^3 = \left(\frac{4}{7}\right)^8 \)

Solution

\begin{align*} &= \left(\frac{4}{7}\right)^5 \times \left(\frac{4}{7}\right)^3 \\ \\ &= \left(\frac{4}{7}\right)^{5+3} \\ \\ &= \left(\frac{4}{7}\right)^8 \end{align*}

Anwer True

(iii) \( \displaystyle \left(-\frac{1}{2}\right)^4 \div \left(-\frac{1}{2}\right)^3 = \left(-\frac{1}{2}\right) \)

Solution

\begin{align*} &= \left(-\frac{1}{2}\right)^4 \div \left(-\frac{1}{2}\right)^3 \\ \\ &= \left(-\frac{1}{2}\right)^{4-3} \\ \\ &= \left(-\frac{1}{2}\right)^{1} \end{align*}

Anwer True

(iv) \( \displaystyle \left(\frac{6}{7}\right)^6 \div \left(\frac{6}{7}\right)^0 = \left(\frac{6}{7}\right)^0\)

Solution

\begin{align*} &= \left(\frac{6}{7}\right)^6 \div \left(\frac{6}{7}\right)^0 \\ \\ &= \left(\frac{6}{7}\right)^{6-0} \\ \\ &= \left(\frac{6}{7}\right)^6 \end{align*}

Anwer False

2. Fill in the blank in each of the following so as to make the statement true.

(i) \( \displaystyle \left(-\frac{2}{3}\right) \times \left(-\frac{2}{3}\right)^3 = \left(-\frac{2}{3}\right)^\boxed{?} \)

Solution

\begin{align*} & = \left(-\frac{2}{3}\right)^1 \times \left(-\frac{2}{3}\right)^3 \\ \\ & = \left(-\frac{2}{3}\right)^\boxed{1 + 3} \\ \\ & = \left(-\frac{2}{3}\right)^\boxed{4} \end{align*}

Anwer \( \displaystyle \left(-\frac{2}{3}\right) \times \left(-\frac{2}{3}\right)^3 = \left(-\frac{2}{3}\right)^{\color{red} \boxed{4}} \)

(ii) \( \displaystyle \left(\frac{4}{5}\right)^\boxed{?} \times \left(\frac{4}{5}\right)^9 = \left(\frac{4}{5}\right)^{11} \)

Solution

\begin{align*} \left(\frac{4}{5}\right)^\boxed{?} & = \left(\frac{4}{5}\right)^{11} \div \left(\frac{4}{5}\right)^9 \\ \\ & = \left(\frac{4}{5}\right)^\boxed{11 - 9} \\ \\ & = \left(\frac{4}{5}\right)^2 \end{align*}

Anwer \( \displaystyle \left(\frac{4}{5}\right)^{\color{red}\boxed{2}} \times \left(\frac{4}{5}\right)^9 = \left(\frac{4}{5}\right)^{11} \)

(iii) \( \displaystyle \left(-\frac{3}{7}\right)^{15} \div \left(-\frac{3}{7}\right)^\boxed{?} = \left(-\frac{3}{7}\right) \)

Solution

\begin{align*} \left(-\frac{3}{7}\right)^\boxed{?} & = \left(-\frac{3}{7}\right)^{15} \div \left(-\frac{3}{7}\right)^1 \\ \\ & = \left(-\frac{3}{7}\right)^\boxed{15 - 1} \\ \\ & = \left(-\frac{3}{7}\right)^{14} \end{align*}

Anwer \( \displaystyle \left(-\frac{3}{7}\right)^{15} \div \left(-\frac{3}{7}\right)^{\color{red}\boxed{14}} = \left(-\frac{3}{7}\right) \)

(iv) \( \displaystyle \left(-\frac{1}{10}\right)^\boxed{?} \div \left(-\frac{1}{10}\right)^8 = \left(-\frac{1}{10}\right)^8 \)

Solution

\begin{align*} \left(-\frac{1}{10}\right)^\boxed{?} &= \left(-\frac{1}{10}\right)^8 \times \left(-\frac{1}{10}\right)^8 \\ \\ &= \left(-\frac{1}{10}\right)^\boxed{8+8} \\ \\ &= \left(-\frac{1}{10}\right)^{16} \end{align*}

Anwer \( \displaystyle \left(-\frac{1}{10}\right)^{\color{red}\boxed{16}} \div \left(-\frac{1}{10}\right)^8 = \left(-\frac{1}{10}\right)^8 \)

(v) \( \displaystyle \left(\frac{2}{9}\right)^6 \div \left(\frac{2}{9}\right)^0 = \left(\frac{2}{9}\right)^\boxed{?} \)

Solution

\begin{align*} \left(\frac{2}{9}\right)^\boxed{?} & = \left(\frac{2}{9}\right)^6 \div \left(\frac{2}{9}\right)^0 \\ \\ & = \left(\frac{2}{9}\right)^\boxed{6 - 0} \\ \\ & = \left(\frac{2}{9}\right)^\boxed{6} \end{align*}

Anwer \( \displaystyle \left(\frac{2}{9}\right)^6 \div \left(\frac{2}{9}\right)^0 = \left(\frac{2}{9}\right)^{\color{red} \boxed{6}} \)

(vi) \( \displaystyle \left(-\frac{12}{13}\right)^2 \times \left(-\frac{12}{13}\right)^\boxed{?} = \left(-\frac{12}{13}\right)^5 \)

Solution

\begin{align*} \left(-\frac{12}{13}\right)^2 \times \left(-\frac{12}{13}\right)^\boxed{?} &= \left(-\frac{12}{13}\right)^5 \\ \\ &= \left(-\frac{12}{13}\right)^5 \div \left(-\frac{12}{13}\right)^2 \\ \\ &= \left(-\frac{12}{13}\right)^\boxed{5 - 2} \\ \\ &= \left(-\frac{12}{13}\right)^\boxed{3} \end{align*}

Anwer \( \displaystyle\left(-\frac{12}{13}\right)^2 \times \left(-\frac{12}{13}\right)^{\color{red} \boxed{3}} = \left(-\frac{12}{13}\right)^5 \)

3. Simplify and express the result in exponential form.

(i) \( \displaystyle \left(-\frac{3}{7}\right)^3 \times \left(-\frac{3}{7}\right)^4 \)

Solution

\begin{align*} &= \left(-\frac{3}{7}\right)^{3 + 4} \\ \\ &= \color{red}\left(-\frac{3}{7}\right)^7 \end{align*}

(ii) \( \displaystyle \left(\frac{11}{12}\right)^{15} \times \left(\frac{11}{12}\right)^5 \times \left(\frac{11}{12}\right)^{10} \)

Solution

\begin{align*} &= \left(\frac{11}{12}\right)^{15+5+10} \\ \\ &= \color{red}\left(\frac{11}{12}\right)^{30} \end{align*}

(iii) \( \displaystyle \left(-\frac{9}{11}\right)^9 \div \left(-\frac{9}{11}\right)^7 \)

Solution

\begin{align*} &= \left(-\frac{9}{11}\right)^{9 - 7} \\ \\ &= \color{red}\left(-\frac{9}{11}\right)^2 \end{align*}

(iv) \( \displaystyle \left(\frac{1}{4}\right)^8 \div \left(\frac{1}{4}\right)^6 \)

Solution

\begin{align*} &= \left(\frac{1}{4}\right)^{8 - 6} \\ \\ &= \color{red}\left(\frac{1}{4}\right)^2 \end{align*}

4. Simplify and express the result as a rational number.

(i) \( \displaystyle \left(\frac{5}{7}\right)^4 \div \left(\frac{5}{7}\right)^2 \)

Solution

\begin{align*} &= \left(\frac{5}{7}\right)^{4 - 2} \\ \\ &= \left(\frac{5}{7}\right)^2 \\ \\ &= \left(\frac{5}{7}\right) \times \left(\frac{5}{7}\right) \\ \\ &= \frac{25}{49} \end{align*}

Answer \( \displaystyle \color{red} \frac{25}{49} \)

(ii) \( \displaystyle \left(-\frac{2}{3}\right)^2 \times \left(-\frac{2}{3}\right)^3 \)

Solution

\begin{align*} &= \left(-\frac{2}{3}\right)^{2 + 3} \\ \\ &= \left(-\frac{2}{3}\right)^5 \\ \\ &= \left(-\frac{2}{3}\right) \times \left(-\frac{2}{3}\right) \times \left(-\frac{2}{3}\right) \times \left(-\frac{2}{3}\right) \times \left(-\frac{2}{3}\right) \\ \\ &= \frac{-32}{243} \end{align*}

Answer \( \displaystyle \color{red} \frac{-32}{243} \)

(iii) \( \displaystyle \left(\frac{3}{4}\right)^3 \times \left(\frac{3}{4}\right)^2 \)

Solution

\begin{align*} &= \left(\frac{3}{4}\right)^{3+2} \\ \\ &= \left(\frac{3}{4}\right)^5 \\ \\ &= \left(\frac{3}{4}\right) \times \left(\frac{3}{4}\right) \times \left(\frac{3}{4}\right) \times \left(\frac{3}{4}\right) \times \left(\frac{3}{4}\right) \\ \\ &= \frac{243}{1024} \end{align*}

Answer \( \displaystyle \color{red} \frac{243}{1024} \)

(iv) \( \displaystyle \left(-\frac{3}{5}\right)^6 \div \left(-\frac{3}{5}\right)^3 \)

Solution

\begin{align*} &= \left(-\frac{3}{5}\right)^{6-3} \\ \\ &= \left(-\frac{3}{5}\right)^3 \\ \\ &= \left(-\frac{3}{5}\right) \times \left(-\frac{3}{5}\right) \times \left(-\frac{3}{5}\right) \\ \\ &= \frac{-27}{125} \end{align*}

Answer \( \displaystyle \color{red} \frac{-27}{125} \)

(v) \( \displaystyle \left(-\frac{1}{10}\right)^4 \times \left(-\frac{1}{10}\right)^2 \)

Solution

\begin{align*} &= \left(-\frac{1}{10}\right)^{4+2} \\ \\ &= \left(-\frac{1}{10}\right)^6 \\ \\ &= \frac{1}{1000000} \end{align*}

Answer \( \displaystyle \color{red} \frac{1}{1000000} \)

5. Evaluate

(i) \( \displaystyle \left(\frac{3}{4}\right)^3 \times \left(\frac{2}{3}\right)^2 \)

Solution

\begin{align*} &= \frac{3}{4} \times \frac{\cancel3^1}{\cancel4_2} \times \frac{\cancel{3^1}}{\cancel4_2} \times \frac{\cancel2^1}{\cancel{3}_1} \times \frac{\cancel2^1}{\cancel3_1} \\ \\ &= \frac{3}{4 \times 2 \times 2} \\ \\ &= \frac{3}{16} \end{align*}

Answer \( \displaystyle \color{red} \frac{3}{16} \)

(ii) \( \displaystyle \left[\left(\frac{1}{3}\right)^6 \div \left(\frac{1}{3}\right)^5\right] \div \left(\frac{1}{3}\right) \)

Solution

\begin{align*} &= \left[\left(\frac{1}{3}\right)^{6-5}\right] \div \left(\frac{1}{3}\right) \\ \\ &= \left(\frac{1}{3}\right)^{1} \div \left(\frac{1}{3}\right) \\ \\ &= \left(\frac{1}{3}\right)^{1-1} \\ \\ &= \left(\frac{1}{3}\right)^0 \\ \\ &= 1 \end{align*}

Answer \( \displaystyle \color{red} 1 \)

(iii) \( \displaystyle ({2^4 \times 2^5}) \div {2^8} \)

Solution

\begin{align*} &= ({2^{4+5}}) \div {2^8} \\ &= {2^{9}} \div {2^8} \\ &= {2^{9-8}} \\ &= 2^{1} \\ &= 2 \end{align*}

Answer \( \displaystyle \color{red} 2 \)

(iv) \( \displaystyle ({4^2 - 3^2}) \div {\left(\frac{1}{7}\right)^2} \)

Solution

\begin{align*} &= ({16 - 9}) \div {\left(\frac{1}{7} \times \frac{1}{7}\right)} \\ \\ &= {7} \div {\frac{1}{49}} \\ \\ &= 7 \times 49 \\ &= 343 \end{align*}

Answer \( \displaystyle \color{red} 343 \)