To simplify [tex]\(\left(\frac{2 x^3}{3 y^2}\right)^4\)[/tex], follow these steps:
1. Expand the power to both the numerator and the denominator:
[tex]\[
\left(\frac{2 x^3}{3 y^2}\right)^4 = \frac{(2 x^3)^4}{(3 y^2)^4}
\][/tex]
2. Simplify the numerator [tex]\((2 x^3)^4\)[/tex]:
When raising a product to a power, you raise each factor to the power:
[tex]\[
(2 x^3)^4 = 2^4 \cdot (x^3)^4
\][/tex]
Compute each part separately:
[tex]\[
2^4 = 16
\][/tex]
For [tex]\((x^3)^4\)[/tex], raise the power of a power by multiplying exponents:
[tex]\[
(x^3)^4 = x^{3 \cdot 4} = x^{12}
\][/tex]
So,
[tex]\[
(2 x^3)^4 = 16 x^{12}
\][/tex]
3. Simplify the denominator [tex]\((3 y^2)^4\)[/tex]:
Similarly, raise each factor to the power:
[tex]\[
(3 y^2)^4 = 3^4 \cdot (y^2)^4
\][/tex]
Compute each part separately:
[tex]\[
3^4 = 81
\][/tex]
For [tex]\((y^2)^4\)[/tex], raise the power of a power by multiplying exponents:
[tex]\[
(y^2)^4 = y^{2 \cdot 4} = y^8
\][/tex]
So,
[tex]\[
(3 y^2)^4 = 81 y^8
\][/tex]
4. Combine the simplified numerator and denominator:
[tex]\[
\frac{(2 x^3)^4}{(3 y^2)^4} = \frac{16 x^{12}}{81 y^8}
\][/tex]
Thus, the simplified result of [tex]\(\left( \frac{2 x^3}{3 y^2} \right)^4\)[/tex] is:
[tex]\[
\frac{16 x^{12}}{81 y^8}
\][/tex]
