To analyze whether there is a difference between doing [tex]\((\sqrt{x})^3\)[/tex] or [tex]\(\sqrt{x^3}\)[/tex] for the expression [tex]\(x^{\frac{3}{2}}\)[/tex], let's go through a detailed explanation with an example.
Let's take [tex]\(x = 4\)[/tex].
1. Calculate [tex]\((\sqrt{x})^3\)[/tex]:
- First, find the square root of [tex]\(x\)[/tex].
[tex]\[
\sqrt{4} = 2
\][/tex]
- Then, raise this result to the power of 3.
[tex]\[
(2)^3 = 8
\][/tex]
Thus, [tex]\((\sqrt{4})^3 = 8\)[/tex].
2. Calculate [tex]\(\sqrt{x^3}\)[/tex]:
- First, raise [tex]\(x\)[/tex] to the power of 3.
[tex]\[
4^3 = 64
\][/tex]
- Then, find the square root of this result.
[tex]\[
\sqrt{64} = 8
\][/tex]
Thus, [tex]\(\sqrt{4^3} = 8\)[/tex].
From both methods, we see that:
[tex]\[
(\sqrt{4})^3 = 8 \quad \text{and} \quad \sqrt{4^3} = 8
\][/tex]
Therefore, there is no difference in the result between [tex]\((\sqrt{x})^3\)[/tex] and [tex]\(\sqrt{x^3}\)[/tex] for the expression [tex]\(x^{\frac{3}{2}}\)[/tex]. Both methods yield the same final answer.
To generalize, for any non-negative [tex]\(x\)[/tex], we have:
[tex]\[
(\sqrt{x})^3 = \sqrt{x^3}
\][/tex]
This can be understood from the property of exponents:
[tex]\[
x^{\frac{3}{2}} = \left(x^{\frac{1}{2}}\right)^3 = \left(x^3\right)^{\frac{1}{2}}
\][/tex]
which confirms that both methods [tex]\((\sqrt{x})^3\)[/tex] and [tex]\(\sqrt{x^3}\)[/tex] will indeed yield the same result.
