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]

C. [tex]\sqrt{5^3}[/tex]

D. [tex](\sqrt[8]{5})^9[/tex]



Answer :

Let's solve the problem step-by-step to determine which expressions are equivalent to \(\left(5^{\frac{1}{8}} \cdot 5^{\frac{3}{8}}\right)^3\).

### Step 1: Simplify the Inside of the Parentheses
First, we need to simplify the expression inside the parentheses \(\left(5^{\frac{1}{8}} \cdot 5^{\frac{3}{8}}\right)\).

We use the property of exponents: when multiplying like bases, we add the exponents. Therefore,

[tex]\[ 5^{\frac{1}{8}} \cdot 5^{\frac{3}{8}} = 5^{\frac{1}{8} + \frac{3}{8}} = 5^{\frac{4}{8}} = 5^{\frac{1}{2}} \][/tex]

### Step 2: Raise the Simplified Expression to the Power of 3
Next, we raise the simplified expression to the power of 3:

[tex]\[ (5^{\frac{1}{2}})^3 \][/tex]

Using the property of exponents \((a^m)^n = a^{m \cdot n}\):

[tex]\[ (5^{\frac{1}{2}})^3 = 5^{\frac{1}{2} \cdot 3} = 5^{\frac{3}{2}} \][/tex]

### Step 3: Compare with the Given Expressions
Now, let's compare this result \(5^{\frac{3}{2}}\) with the given expressions one by one to see which are equivalent.

1. \(5^{\frac{3}{2}}\):
- This is exactly what we derived, so it is equivalent.

2. \(5^{\frac{9}{8}}\):
- This is different from \(5^{\frac{3}{2}}\).

3. \(\sqrt{5^3}\):
- This expression can be simplified. Recall that the square root of a number is the same as raising it to the power of \( \frac{1}{2} \):

[tex]\[ \sqrt{5^3} = (5^3)^{\frac{1}{2}} = 5^{3 \cdot \frac{1}{2}} = 5^{\frac{3}{2}} \][/tex]

- So, \(\sqrt{5^3}\) is equivalent to \(5^{\frac{3}{2}}\).

4. \((\sqrt[8]{5})^9\):
- This expression can be simplified. Recall that the 8th root of a number is the same as raising it to the power of \(\frac{1}{8}\):

[tex]\[ (\sqrt[8]{5})^9 = (5^{\frac{1}{8}})^9 = 5^{\frac{1}{8} \cdot 9} = 5^{\frac{9}{8}} \][/tex]

- This is different from \(5^{\frac{3}{2}}\).

### Conclusion
The expressions equivalent to \(\left(5^{\frac{1}{8}} \cdot 5^{ \frac{3}{8}}\right)^3\) are:

[tex]\[ 5^{\frac{3}{2}} \quad \text{and} \quad \sqrt{5^3} \][/tex]