Simplify and express the final result using positive exponents. (\frac{4a

tabita57i 2021-09-26 Answered
Simplify and express the final result using positive exponents.
\(\displaystyle{\left({\frac{{{4}{a}^{{-{2}}}}}{{{3}{b}^{{-{2}}}}}}\right)}^{{-{2}}}\)

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Expert Answer

faldduE
Answered 2021-09-27 Author has 14700 answers
We have to simplify the expression:
\(\displaystyle{\left({\frac{{{4}{a}^{{-{2}}}}}{{{3}{b}^{{-{2}}}}}}\right)}^{{-{2}}}\)
We know the exponents rule,
\(\displaystyle{\left({\frac{{{a}}}{{{b}}}}\right)}^{{m}}={\frac{{{a}^{{m}}}}{{{b}^{{m}}}}}\)
\(\displaystyle{\left({a}^{{m}}\right)}^{{n}}={a}^{{{n}\times{n}}}\)
\(\displaystyle{a}^{{-{m}}}={\frac{{{1}}}{{{a}^{{m}}}}}\)
Applying above rule for the given expression, we get
\(\displaystyle{\left({\frac{{{4}{a}^{{-{2}}}}}{{{3}{b}^{{-{2}}}}}}\right)}^{{-{2}}}={\left({\frac{{{4}{a}^{{-{2}}}}}{{{3}{b}^{{-{2}}}}}}\right)}^{{-{2}}}\)
\(\displaystyle={\frac{{{\left({4}{a}^{{-{2}}}\right)}^{{-{2}}}}}{{{\left({3}{b}^{{-{2}}}\right)}^{{-{2}}}}}}\)
\(\displaystyle={\frac{{{\left({4}\right)}^{{-{2}}}{\left({a}^{{-{2}}}\right)}^{{-{2}}}}}{{{\left({3}\right)}^{{-{2}}}{\left({b}^{{-{2}}}\right)}^{{-{2}}}}}}\)
\(\displaystyle={\frac{{{\frac{{{1}}}{{{4}^{{2}}}}}{a}^{{{\left(-{2}\right)}\times{\left(-{2}\right)}}}}}{{{\frac{{{1}}}{{{3}^{{2}}}}}{b}^{{{\left(-{2}\right)}\times{\left(-{2}\right)}}}}}}\)
\(\displaystyle={\frac{{{\frac{{{1}}}{{{16}}}}{a}^{{4}}}}{{{\frac{{{1}}}{{{9}}}}{b}^{{4}}}}}\)
\(\displaystyle={\frac{{{9}{a}^{{4}}}}{{{16}{b}^{{4}}}}}\)
Hence, simplified form of the expression is \(\displaystyle{\frac{{{9}{a}^{{4}}}}{{{16}{b}^{{4}}}}}\)
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