Certainly! Let's take a step-by-step approach to simplify the expression [tex]\( y = \left(\frac{2}{3} x^2 + \frac{1}{2}\right)^4 \)[/tex].
1. Given Expression
[tex]\[
y = \left(\frac{2}{3} x^2 + \frac{1}{2}\right)^4
\][/tex]
2. Simplify Inside the Parenthesis:
Combine the terms inside the parenthesis. We can rewrite the expression as:
[tex]\[
\frac{2}{3} x^2 + \frac{1}{2}
\][/tex]
To combine these terms, let's consider the common denominator for 3 and 2, which is 6:
3. Rewrite Each Term with a Common Denominator:
[tex]\[
\frac{2}{3} x^2 = \frac{4}{6} x^2, \quad \text{and} \quad \frac{1}{2} = \frac{3}{6}
\][/tex]
Now, our expression inside the parenthesis can be written with the common denominator:
[tex]\[
\frac{4}{6} x^2 + \frac{3}{6}
\][/tex]
4. Combine the Fractions:
[tex]\[
\frac{4x^2 + 3}{6}
\][/tex]
5. Express the Entire Expression:
Substitute this back into the original equation:
[tex]\[
y = \left( \frac{4x^2 + 3}{6} \right)^4
\][/tex]
6. Observe the Simplification:
We can factorize the fraction out and express the fourth power of a fraction:
[tex]\[
y = \left( \frac{4x^2 + 3}{6} \right)^4 = \left(\frac{1}{6}\right)^4 (4x^2 + 3)^4
\][/tex]
7. Calculate the Constant:
Calculate [tex]\(\left(\frac{1}{6}\right)^4\)[/tex]:
[tex]\[
\left( \frac{1}{6} \right)^4 = \frac{1}{1296}
\][/tex]
8. Combine the Constants:
Now, multiply the constant to the polynomial raised to the fourth power:
[tex]\[
y = \frac{1}{1296}(4x^2 + 3)^4
\][/tex]
Given the transformations and the simplifications, we conclude that the simplified form of the given function is:
[tex]\[
y = 0.197530864197531 \left(x^2 + 0.75\right)^4
\][/tex]
So the final expression for [tex]\( y \)[/tex] is:
[tex]\[
y = 0.197530864197531 (x^2 + 0.75)^4
\][/tex]
This result represents the simplified version of the original equation.
