Ex.12.1 Q4 Exponents and Powers Solution - NCERT Maths Class 8

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Question

Evaluate

(i)\(\begin{align}\, \frac{{{8^{ - 1}} \times {5^3}}}{{{2^{ - 4}}}}\end{align}\)

(ii)\(\, ({5^{ - 1}} \times {2^{ - 1}}) \times {6^{ - 1}}\)

Text Solution

(i) Evaluate\(\begin{align}\, \frac{{{8^{ - 1}} \times {5^3}}}{{{2^{ - 4}}}}\end{align}\)

What is known?

Expression in exponential form

What is unknown?

Value of the expression

Reasoning:

\[\begin{align}{a^{ - m}}& = \frac{1}{{{a^m}}}\\{a^m} \div {a^n} &= {a^{m - n}}\end{align}\]

Steps:

\[\begin{align}&\frac{{{8^{ - 1}} \times {5^3}}}{{{2^{ - 4}}}}\\&= \frac{{{2^4} \times {5^3}}}{{{8^1}}} \quad \left[ {{a^{ - {{m}}}} = \frac{1}{{{a^{{m}}}}}} \right]\\&= \frac{{{2^4} \times {5^3}}}{{{2^3}}}\\&= {2^{4 - 3}} \times {5^3} \quad \left[ {{a^{{m}}} \div {a^{{n}}} = {a^{{{m - n}}}}} \right]\\&= 2 \times 125\\&= 250\end{align}\]

(ii) Evaluate \(\quad ({5^{ - 1}} \times {2^{ - 1}}) \times {6^{ - 1}}\)

What is known?

Expression in exponential form

What is unknown?

Value of the expression

Reasoning:

\[{a^{{m}}} \times {b^{{m}}} = {(ab)^{{m}}}\]

Steps:

\[\begin{align}&({5^{ - 1}} \times {2^{ - 1}}) \times {6^{ - 1}} \\&= {10^{ - 1}} \times {6^{ - 1}} \\&= {(10 \times 6)^{ - 1}} \\&\qquad \left[ {\because \,{a^m} \times {b^m} = {{\left( {ab} \right)}^m}} \right] \\&= {(60)^{ - 1}} \\&= \frac{1}{{60}} \\\end{align} \]

Related problems:

(i) \(({7^{ - 2}} \times {14^{ - 2}}) \times {3^{ - 1}}\)

(ii) \(\begin{align}\frac{{{2^{ - 2}} \times {5^2}}}{{{8^{ - 3}}}}\end{align}\)

  
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