To simplify the expression
[tex]\[
\frac{(2 y^5)^5}{2 y^5}
\][/tex]
we need to follow these steps:
1. Simplify the numerator:
The numerator is [tex]\((2 y^5)^5\)[/tex]. We need to apply the power rule [tex]\((a b)^n = a^n b^n\)[/tex]:
[tex]\[
(2 y^5)^5 = 2^5 \cdot (y^5)^5 = 2^5 \cdot y^{5 \cdot 5} = 2^5 \cdot y^{25}
\][/tex]
2. Simplify the denominator:
The denominator is [tex]\(2 y^5\)[/tex], which remains as it is.
3. Divide the simplified numerator by the simplified denominator:
[tex]\[
\frac{2^5 \cdot y^{25}}{2 y^5}
\][/tex]
We can separate this fraction into two parts: one involving the coefficients and one involving the powers of [tex]\(y\)[/tex]:
[tex]\[
\frac{2^5}{2} \times \frac{y^{25}}{y^5}
\][/tex]
4. Simplify each part:
- For the coefficients:
[tex]\[
\frac{2^5}{2} = \frac{32}{2} = 16
\][/tex]
- For the powers of [tex]\(y\)[/tex], use the property [tex]\(\frac{y^a}{y^b} = y^{a-b}\)[/tex]:
[tex]\[
\frac{y^{25}}{y^5} = y^{25-5} = y^{20}
\][/tex]
5. Combine the results:
[tex]\[
\frac{(2 y^5)^5}{2 y^5} = 16 \cdot y^{20}
\][/tex]
Therefore, the coefficient [tex]\(c\)[/tex] is [tex]\(16\)[/tex] and the exponent [tex]\(e\)[/tex] of [tex]\(y\)[/tex] is [tex]\(20\)[/tex].
So, the expression [tex]\(\frac{(2 y^5)^5}{2 y^5}\)[/tex] equals [tex]\(16 y^{20}\)[/tex] where the coefficient [tex]\(c\)[/tex] is [tex]\(16\)[/tex] and the exponent [tex]\(e\)[/tex] is [tex]\(20\)[/tex].
