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Rational Exponents

Which expression(s) are equivalent to [tex]\left(5^{\frac{1}{8}} \cdot 5^{\frac{3}{8}}\right)^3[/tex]?

A. [tex]5^{\frac{3}{2}}[/tex]

B. [tex]5^{\frac{9}{8}}[/tex]


Sagot :

Certainly! Let's go through the detailed, step-by-step solution to determine which expressions are equivalent to [tex]\(\left(5^{\frac{1}{8}} \cdot 5^{\frac{3}{8}}\right)^3\)[/tex].

1. Simplify the expression inside the parentheses:
[tex]\[ 5^{\frac{1}{8}} \cdot 5^{\frac{3}{8}} \][/tex]

When multiplying expressions with the same base, we add the exponents:
[tex]\[ 5^{\frac{1}{8}} \cdot 5^{\frac{3}{8}} = 5^{\frac{1}{8} + \frac{3}{8}} \][/tex]

2. Add the exponents:
[tex]\[ \frac{1}{8} + \frac{3}{8} = \frac{4}{8} = \frac{1}{2} \][/tex]

So, the expression simplifies to:
[tex]\[ 5^{\frac{1}{2}} \][/tex]

3. Now consider the entire expression:
[tex]\[ \left(5^{\frac{1}{2}}\right)^3 \][/tex]

When raising a power to another power, we multiply the exponents:
[tex]\[ \left(5^{\frac{1}{2}}\right)^3 = 5^{\frac{1}{2} \cdot 3} = 5^{\frac{3}{2}} \][/tex]

Thus, the simplified expression is:
[tex]\[ 5^{\frac{3}{2}} \][/tex]

We can now compare this with the given options:

- Option 1: [tex]\(5^{\frac{3}{2}}\)[/tex] is identical to our simplified result, so it is equivalent.

- Option 2: [tex]\(5^{\frac{9}{8}}\)[/tex] is not equivalent to our simplified result, as [tex]\(\frac{9}{8}\)[/tex] is different from [tex]\(\frac{3}{2}\)[/tex].

Therefore, the expression [tex]\(\left(5^{\frac{1}{8}} \cdot 5^{\frac{3}{8}}\right)^3\)[/tex] is equivalent to [tex]\(5^{\frac{3}{2}}\)[/tex].
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