To solve the problem of evaluating the product
[tex]\[
\sqrt[4]{7} \cdot \sqrt[4]{7} \cdot \sqrt[4]{7} \cdot \sqrt[4]{7},
\][/tex]
we need to understand how to work with roots and exponents.
First, let's express [tex]\(\sqrt[4]{7}\)[/tex] using exponents. The fourth root of 7 can be written as:
[tex]\[
\sqrt[4]{7} = 7^{\frac{1}{4}}.
\][/tex]
Next, consider the given product:
[tex]\[
\sqrt[4]{7} \cdot \sqrt[4]{7} \cdot \sqrt[4]{7} \cdot \sqrt[4]{7}.
\][/tex]
Using the exponent form, this is:
[tex]\[
7^{\frac{1}{4}} \cdot 7^{\frac{1}{4}} \cdot 7^{\frac{1}{4}} \cdot 7^{\frac{1}{4}}.
\][/tex]
We can simplify the product of these exponentials by using the property of exponents that states [tex]\(a^m \cdot a^n = a^{m+n}\)[/tex]. Applying this property, we get:
[tex]\[
7^{\frac{1}{4}} \cdot 7^{\frac{1}{4}} \cdot 7^{\frac{1}{4}} \cdot 7^{\frac{1}{4}} = 7^{\frac{1}{4} + \frac{1}{4} + \frac{1}{4} + \frac{1}{4}}.
\][/tex]
Next, add the exponents together:
[tex]\[
\frac{1}{4} + \frac{1}{4} + \frac{1}{4} + \frac{1}{4} = 1.
\][/tex]
Therefore, we have:
[tex]\[
7^{\frac{1}{4} + \frac{1}{4} + \frac{1}{4} + \frac{1}{4}} = 7^1.
\][/tex]
And simplifying the exponential, we obtain:
[tex]\[
7^1 = 7.
\][/tex]
So, the product of [tex]\(\sqrt[4]{7} \cdot \sqrt[4]{7} \cdot \sqrt[4]{7} \cdot \sqrt[4]{7}\)[/tex] is:
[tex]\[
\boxed{7}.
\][/tex]
