Question

Use the power rules for exponents to simplify the expression. Give your answer in exponential form.

left parenthesis StartFraction 4 a squared b Superscript 7 Over c Superscript 5 EndFraction right parenthesis squared
​(cnot equals​0)

Answer 1

To simplify the expression \(\left(\frac{4a^2b^7}{c^5}\right)^2\) using the power rules for exponents, we will apply the power of a quotient rule and the power of a product rule as follows:

1. **Power of a Quotient Rule**: \(\left(\frac{x}{y}\right)^n = \frac{x^n}{y^n}\)
2. **Power of a Product Rule**: \((xy)^n = x^n y^n\)

Now, applying these rules to the given expression:

\[
\left(\frac{4a^2b^7}{c^5}\right)^2 = \frac{(4)^2(a^2)^2(b^7)^2}{(c^5)^2}
\]

Now we simplify each component:

- \((4)^2 = 16\)
- \((a^2)^2 = a^{2 \cdot 2} = a^4\)
- \((b^7)^2 = b^{7 \cdot 2} = b^{14}\)
- \((c^5)^2 = c^{5 \cdot 2} = c^{10}\)

Putting it all together, we have:

\[
\frac{16a^4b^{14}}{c^{10}}
\]

Thus, the simplified expression in exponential form is:

\[
\frac{16a^4b^{14}}{c^{10}}
\]