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

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

• Questions are typically answered in as fast as 30 minutes

### Plainmath recommends

• Get a detailed answer even on the hardest topics.
• Ask an expert for a step-by-step guidance to learn to do it yourself.

faldduE
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 $$\displaystyle{\frac{{{9}{a}^{{4}}}}{{{16}{b}^{{4}}}}}$$