Answer :
Let's break down the expression [tex]\(\sqrt[4]{5^5} \cdot \sqrt[6]{5^5}\)[/tex] to understand and simplify it.
### Step 1: Express each component with fractional exponents
The expression [tex]\(\sqrt[4]{5^5}\)[/tex] can be rewritten as [tex]\((5^5)^{1/4}\)[/tex]. Similarly, [tex]\(\sqrt[6]{5^5}\)[/tex] can be rewritten as [tex]\((5^5)^{1/6}\)[/tex].
### Step 2: Use the property of exponents
When multiplying terms with the same base, you add the exponents. Thus, the expression becomes:
[tex]\[
5^{5 \cdot \frac{1}{4}} \cdot 5^{5 \cdot \frac{1}{6}} = 5^{(5 \cdot \frac{1}{4}) + (5 \cdot \frac{1}{6})}
\][/tex]
### Step 3: Calculate each part of the exponent
First, calculate [tex]\(5 \times \frac{1}{4}\)[/tex]:
[tex]\[
5 \times \frac{1}{4} = \frac{5}{4} = 1.25
\][/tex]
Next, calculate [tex]\(5 \times \frac{1}{6}\)[/tex]:
[tex]\[
5 \times \frac{1}{6} = \frac{5}{6} \approx 0.8333
\][/tex]
### Step 4: Add the exponents
Now, add the calculated exponents:
[tex]\[
1.25 + 0.8333 = 2.0833
\][/tex]
### Step 5: The simplified expression
The simplified expression with the combined exponent is:
[tex]\[
5^{2.0833}
\][/tex]
Therefore, the expression [tex]\(\sqrt[4]{5^5} \cdot \sqrt[6]{5^5}\)[/tex] simplifies to approximately [tex]\(5^{2.0833}\)[/tex].
### Step 1: Express each component with fractional exponents
The expression [tex]\(\sqrt[4]{5^5}\)[/tex] can be rewritten as [tex]\((5^5)^{1/4}\)[/tex]. Similarly, [tex]\(\sqrt[6]{5^5}\)[/tex] can be rewritten as [tex]\((5^5)^{1/6}\)[/tex].
### Step 2: Use the property of exponents
When multiplying terms with the same base, you add the exponents. Thus, the expression becomes:
[tex]\[
5^{5 \cdot \frac{1}{4}} \cdot 5^{5 \cdot \frac{1}{6}} = 5^{(5 \cdot \frac{1}{4}) + (5 \cdot \frac{1}{6})}
\][/tex]
### Step 3: Calculate each part of the exponent
First, calculate [tex]\(5 \times \frac{1}{4}\)[/tex]:
[tex]\[
5 \times \frac{1}{4} = \frac{5}{4} = 1.25
\][/tex]
Next, calculate [tex]\(5 \times \frac{1}{6}\)[/tex]:
[tex]\[
5 \times \frac{1}{6} = \frac{5}{6} \approx 0.8333
\][/tex]
### Step 4: Add the exponents
Now, add the calculated exponents:
[tex]\[
1.25 + 0.8333 = 2.0833
\][/tex]
### Step 5: The simplified expression
The simplified expression with the combined exponent is:
[tex]\[
5^{2.0833}
\][/tex]
Therefore, the expression [tex]\(\sqrt[4]{5^5} \cdot \sqrt[6]{5^5}\)[/tex] simplifies to approximately [tex]\(5^{2.0833}\)[/tex].