Sure, let's evaluate the expression [tex]\(100^{3 / 2}\)[/tex] step by step.
1. Understand the Meaning of Exponentiation:
To evaluate [tex]\(100^{3 / 2}\)[/tex], we should recognize that the exponent [tex]\(\frac{3}{2}\)[/tex] can be interpreted using properties of exponents. Essentially, [tex]\(100^{3 / 2}\)[/tex] means that we first take the square root of 100, and then raise the result to the power of 3, or alternatively, raise 100 to the power of 3 and then take the square root of the result.
2. Breaking Down the Exponent:
Let's rewrite [tex]\(100^{3 / 2}\)[/tex] for clarity:
[tex]\[
100^{3 / 2} = (100^{1 / 2})^3 = (\sqrt{100})^3
\][/tex]
3. Evaluate the Square Root:
We know that the square root of 100 is:
[tex]\[
\sqrt{100} = 10
\][/tex]
4. Raise the Result to the Power of 3:
Now we need to take our result from the previous step (which is 10) and raise it to the power of 3:
[tex]\[
10^3 = 10 \times 10 \times 10 = 1000
\][/tex]
So, the value of [tex]\(100^{3 / 2}\)[/tex] is:
[tex]\[
1000
\][/tex]
Thus, the real solution to [tex]\(100^{3 / 2}\)[/tex] is [tex]\(1000.0\)[/tex].
