20. Which of the following is equivalent to [tex]25^{\frac{7}{2}}[/tex]?

A. [tex]25^{-\frac{2}{7}}[/tex]
B. [tex]25^{-\frac{7}{2}}[/tex]
C. [tex]\sqrt[7]{25^2}[/tex]
D. [tex]\sqrt{25^7}[/tex]



Answer :

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]