Sure! Let's go through the simplification of each of the given expressions step by step.
### Part (a): [tex]\(\sqrt{25 a^4 b^8 - 9 a^4 b^8}\)[/tex]
1. Combine like terms inside the square root:
[tex]\[
25 a^4 b^8 - 9 a^4 b^8 = (25 - 9) a^4 b^8 = 16 a^4 b^8
\][/tex]
2. Simplify the square root:
[tex]\[
\sqrt{16 a^4 b^8} = \sqrt{16} \cdot \sqrt{a^4} \cdot \sqrt{b^8}
\][/tex]
3. Evaluate each square root separately:
[tex]\[
\sqrt{16} = 4, \quad \sqrt{a^4} = a^2, \quad \sqrt{b^8} = b^4
\][/tex]
4. Combine the results:
[tex]\[
\sqrt{16 a^4 b^8} = 4a^2 b^4
\][/tex]
So, the simplified form of [tex]\(\sqrt{25 a^4 b^8 - 9 a^4 b^8}\)[/tex] is:
[tex]\[
4a^2 b^4
\][/tex]
### Part (b): [tex]\(\sqrt{8 y \times 2 y}\)[/tex]
1. Multiply inside the square root:
[tex]\[
8 y \times 2 y = 16 y^2
\][/tex]
2. Simplify the square root:
[tex]\[
\sqrt{16 y^2} = \sqrt{16} \cdot \sqrt{y^2}
\][/tex]
3. Evaluate each square root separately:
[tex]\[
\sqrt{16} = 4, \quad \sqrt{y^2} = y
\][/tex]
4. Combine the results:
[tex]\[
\sqrt{16 y^2} = 4y
\][/tex]
So, the simplified form of [tex]\(\sqrt{8 y \times 2 y}\)[/tex] is:
[tex]\[
4y
\][/tex]
### Part (c): [tex]\(c \cdot \frac{\sqrt{16 x^2}}{2 x}\)[/tex]
1. Simplify the square root in the numerator:
[tex]\[
\sqrt{16 x^2} = \sqrt{16} \cdot \sqrt{x^2}
\][/tex]
2. Evaluate each square root separately:
[tex]\[
\sqrt{16} = 4, \quad \sqrt{x^2} = x
\][/tex]
3. Combine the results in the numerator:
[tex]\[
\sqrt{16 x^2} = 4x
\][/tex]
4. Substitute back into the original expression:
[tex]\[
c \cdot \frac{4x}{2x}
\][/tex]
5. Simplify the fraction:
[tex]\[
\frac{4x}{2x} = 2
\][/tex]
6. Combine the results with [tex]\(c\)[/tex]:
[tex]\[
c \cdot 2 = 2c
\][/tex]
So, the simplified form of [tex]\(c \cdot \frac{\sqrt{16 x^2}}{2 x}\)[/tex] is:
[tex]\[
2c
\][/tex]
