Ex.12.1 Q2 Exponents and Powers - NCERT Maths Class 8

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Question

Simplify and express the result in power notation with positive exponent.

(i) \(\begin{align} {( - 4)^5} \div {( - 4)^8}\end{align}\)

(ii) \(\begin{align} {\left( {\frac{1}{{{2^3}}}} \right)^2}\end{align}\)

(iii) \(\begin{align} {( - 3)^4} \times {\left( {\frac{5}{3}} \right)^4}\end{align}\)

(iv) \(\begin{align}\,\,({{3}^{-7}}\div {{3}^{-10}})\times {{3}^{-5}}\end{align}\)

(v) \(\begin{align} {2^{ - 3}} \times {( - 7)^{ - 3}}\end{align}\)

Text Solution

(i) Evaluate \(\begin{align}{( - 4)^5} \div {( - 4)^8}\end{align}\)

What is known?

Expression in exponential form

What is unknown?

Result in power notation with positive exponent

Reasoning:

As we know \(\begin{align}\frac{{{a^m}}}{{{a^n}}} = {a^{m - n}}\end{align}\) where \(\begin{align}m\end{align}\) & \(\begin{align}n\end{align}\) are integers.

Steps:

\[\begin{align}{{(-4)}^{5}}\div {{(-4)}^{8}}& =\frac{{{(-4)}^{5}}}{{{(-4)}^{8}}}={{(-4)}^{5-8}} \\ {{(-4)}^{-3}}&={{\left( \frac{1}{-4} \right)}^{3}}\end{align}\]

(ii) Evaluate \(\begin{align}\quad {\left( {\frac{1}{{{2^3}}}} \right)^2}\end{align}\)

What is known?

Expression in exponential form

What is unknown?

Result in power notation with positive exponent

Reasoning:

As we know for any non-zero integer a,\(\begin{align}{({a^{{m}}})^{{n}}} = {a^{{{mn}}}}\end{align}\)

Steps:

\[\begin{align}{\left( {\frac{1}{{{2^3}}}} \right)^2}&= \frac{1}{{{2^6}}}\end{align}\]

(iii) Evaluate \(\begin{align} {( - 3)^4} \times {\left( {\frac{5}{3}} \right)^4}\end{align}\)

What is known?

Expression in exponential form

What is unknown?

Result in power notation with positive exponent

Reasoning:

We know that \(\begin{align}{a^{{m}}} \times {b^{{m}}} = {(ab)^{{m}}}\end{align}\) where \(a\) & \(b\) are non-zero integers and \(m\) is any integer

Steps:

\[\begin{align}& {{(-3)}^{4}}\times {{\left( \frac{5}{3} \right)}^{4}} \\& {{(-1\times 3)}^{4}}\times \frac{{{5}^{4}}}{{{3}^{4}}} \\& {{(-1)}^{4}}\times {{{\not\!{3}}}^{4}}\times \frac{{{5}^{4}}}{{{{\not\!{3}}}^{4}}} \\& {{(-1)}^{4}}\times {{5}^{4}}={{5}^{4}}\qquad[\because {{(-1)}^{4}}=1] \\\end{align}\]

(iv) Evaluate \(\begin{align}\,\,({{3}^{-7}}\div {{3}^{-10}})\times {{3}^{-5}}\end{align}\)

What is known?

Expression in exponential form

What is unknown?

Result in power notation with positive exponent

Reasoning:

We know \(\begin{align}\frac{{{{{a}}^{{m}}}}}{{{{{a}}^{{n}}}}}{{ = }}{{{a}}^{{{m - n}}}}\end{align}\) & \(\begin{align}{{{a}}^{{m}}}{{ \times }}{{{a}}^{{n}}}{{ = }}{{{a}}^{{{m + n}}}}\end{align}\)

Steps:

\[\begin{align}({3^{ - 7}} \! \div \! {3^{ - 10}}) \! \times \! {3^{ - 5}} & \! = \! ({3^{ - 7 - ( - 10)}}) \! \times \! {3^{ - 5}}\\& \! = \! ({3^{ - 7 + 10}}) \! \times \! {3^{ - 5}}\\& \! = \! {3^3} \! + \! ({3^{ - 5}})\\& \! = \! {3^{3 + ( - 5)}}\\& \! = \! {3^{ - 2}}\\& \! = \! \frac{1}{{{3}^{2}}}\end{align}\]

(v) Evaluate \(\quad {2^{ - 3}} \times {( - 7)^{ - 3}}\)

What is known?

Expression in exponential form

What is unknown?

Result in power notation with positive exponent

Reasoning:

As we know \(\begin{align} {{{a}}^{{m}}}\,{{ \times }}\,{{{b}}^{{m}}}{{ = }}{\left( {{{ab}}} \right)^{{m}}}\end{align}\)

Steps:

\[\begin{align}{2^{ - 3}} \! \times \! {( - 7)^{ - 3}} & \! = \! {[2 \! \times \! ( - 7)]^{ - 3}}\\& \! = \! [ - 14]^{ - 3} \\ \text{Since }&[{a^{ - m}} \! = \! \frac{1}{{{a^m}}}]\\& \! = \! {\left( {\frac{{ - 1}}{{14}}} \right)^3}\\\end{align}\]

  
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