The expression
[tex]\[ \frac{\left(2 y^5\right)^5}{2 y^5} \][/tex]
equals [tex]\( c y^e \)[/tex] where:

The coefficient [tex]\( c \)[/tex] is [tex]\(\square\)[/tex] , the exponent [tex]\( e \)[/tex] of [tex]\( y \)[/tex] is [tex]\(\square\)[/tex].



Answer :

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].