Let's work through the problem step-by-step to understand the equivalent form of [tex]\(25^{\frac{7}{2}}\)[/tex].
### Step 1: Understand the Exponent Notation
We are given the expression [tex]\(25^{\frac{7}{2}}\)[/tex].
### Step 2: Rewrite the Exponent Notation
The fraction exponent [tex]\(\frac{7}{2}\)[/tex] can be interpreted as taking the square root (since the denominator is 2) and raising it to the power of 7 (since the numerator is 7). Therefore, we can rewrite the expression as:
[tex]\[ 25^{\frac{7}{2}} = (25^{\frac{1}{2}})^7 \][/tex]
### Step 3: Simplify Intermediate Expressions
Now, let's simplify [tex]\(25^{\frac{1}{2}}\)[/tex]. The expression [tex]\( 25^{\frac{1}{2}} \)[/tex] is equivalent to the square root of 25:
[tex]\[ 25^{\frac{1}{2}} = \sqrt{25} \][/tex]
Since [tex]\(25 = 5^2\)[/tex], the square root of 25 is:
[tex]\[ \sqrt{25} = 5 \][/tex]
### Step 4: Raise to the Power of 7
Now, substitute back the simplified value:
[tex]\[ (25^{\frac{1}{2}})^7 = (5)^7 \][/tex]
### Step 5: Combine the Results
Thus, we have:
[tex]\[ 25^{\frac{7}{2}} = 5^7 \][/tex]
### Step 6: Alternative Representation
An alternative way to express the value without calculating [tex]\(5^7\)[/tex] directly is to recognize the equivalent form:
[tex]\[ 25^{\frac{7}{2}} = \sqrt{25^7} \][/tex]
This alternative form directly connects back to our original concept of fraction exponents, so it maintains the equivalence.
### Conclusion
Given the options:
- [tex]\(25-\frac{2}{7}\)[/tex]
- [tex]\(25-\frac{7}{2}\)[/tex]
- [tex]\(\sqrt[7]{25^2}\)[/tex]
- [tex]\(\sqrt{25^7}\)[/tex]
The expression equivalent to [tex]\(25^{\frac{7}{2}}\)[/tex] is:
[tex]\[
\boxed{\sqrt{25^7}}
\][/tex]
