Absolutely, let's simplify each of these expressions step-by-step:
### a) [tex]\((5y^3 - 9y^3)^2\)[/tex]
1. Combine the like terms inside the parentheses:
[tex]\[
5y^3 - 9y^3 = -4y^3
\][/tex]
2. Square the resulting expression:
[tex]\[
(-4y^3)^2 = 16y^6
\][/tex]
Thus, the simplified form is:
[tex]\[ \left(5y^3 - 9y^3\right)^2 = 16y^6 \][/tex]
### b) [tex]\((-25x + 16x)^3\)[/tex]
1. Combine the like terms inside the parentheses:
[tex]\[
-25x + 16x = -9x
\][/tex]
2. Cube the resulting expression:
[tex]\[
(-9x)^3 = -729x^3
\][/tex]
Thus, the simplified form is:
[tex]\[ (-25x + 16x)^3 = -729x^3 \][/tex]
### c) [tex]\(\sqrt{16x^4} - \sqrt[3]{-27x^6}\)[/tex]
1. Simplify the square root part:
[tex]\[
\sqrt{16x^4} = 4x^2
\][/tex]
2. Simplify the cube root part:
[tex]\[
\sqrt[3]{-27x^6} = -3x^2
\][/tex]
3. Subtract the cube root result from the square root result:
[tex]\[
4x^2 - (-3x^2) = 4x^2 + 3x^2 = 7x^2
\][/tex]
Thus, the simplified form is:
[tex]\[ \sqrt{16x^4} - \sqrt[3]{-27x^6} = 7x^2 \][/tex]
### d) [tex]\(\sqrt[3]{36y^3 - 9y^3}\)[/tex]
1. Combine the like terms inside the cube root:
[tex]\[
36y^3 - 9y^3 = 27y^3
\][/tex]
2. Simplify the cube root:
[tex]\[
\sqrt[3]{27y^3} = 3y
\][/tex]
Thus, the simplified form is:
[tex]\[ \sqrt[3]{36y^3 - 9y^3} = 3y \][/tex]
To summarize, the simplified forms are:
- [tex]\((5y^3 - 9y^3)^2 = 16y^6\)[/tex]
- [tex]\((-25x + 16x)^3 = -729x^3\)[/tex]
- [tex]\(\sqrt{16x^4} - \sqrt[3]{-27x^6} = 7x^2\)[/tex]
- [tex]\(\sqrt[3]{36y^3 - 9y^3} = 3y\)[/tex]
With example values [tex]\(x = 2\)[/tex] and [tex]\(y = 3\)[/tex], we get:
- For (a): [tex]\(16 \times (3^6) = 16 \times 729 = 11664\)[/tex]
- For (b): [tex]\(-729 \times (2^3) = -729 \times 8 = -5832\)[/tex]
- For (c): [tex]\(7 \times (2^2) = 7 \times 4 = 28\)[/tex]
- For (d): [tex]\(3 \times 3 = 9\)[/tex]
Thus, the results are:
[tex]\[ (11664, -5832, 28, 9) \][/tex]
