### Solution Explanation
These three students, Jerry, Beth, and Adam, all attempt to solve the equation [tex]\( 25^{2x} = 125^{1/3} \)[/tex] but each takes a different approach. Let's analyze their methods and see who got it right.
### 1. Jerry's Approach
#### Steps:
1. Jerry starts with the equation:
[tex]\[
25^{2x} = 125^{1/3}
\][/tex]
2. He converts the bases 25 and 125 to powers of 5:
[tex]\[
25 = 5^2 \quad \text{and} \quad 125 = 5^3
\][/tex]
3. Therefore, the equation becomes:
[tex]\[
(5^2)^{2x} = (5^3)^{1/3}
\][/tex]
4. Using the exponent rules [tex]\((a^m)^n = a^{mn}\)[/tex], Jerry further simplifies:
[tex]\[
5^{4x} = 5^1
\][/tex]
5. Since the bases are the same, he can equate the exponents:
[tex]\[
4x = 1
\][/tex]
6. Solving for [tex]\(x\)[/tex]:
[tex]\[
x = \frac{1}{4}
\][/tex]
#### Conclusion:
Jerry's steps are logically correct, and he arrives at the correct solution [tex]\( x = \frac{1}{4} \)[/tex].
### 2. Beth's Approach
#### Steps:
1. Beth starts with the same equation:
[tex]\[
25^{2x} = 125^{1/3}
\][/tex]
2. Beth attempts to convert the numbers directly without maintaining proper exponent rules:
[tex]\[
25^{2x} = 5^{4x}
\][/tex]
3. She then incorrectly manipulates the right side of the equation to:
[tex]\[
125^{6x} = 125^{1/3}
\][/tex]
4. Equating exponents from the wrong base assumptions, we get:
[tex]\[
6x = \frac{1}{3}
\][/tex]
5. Solving for [tex]\(x\)[/tex]:
[tex]\[
x = \frac{1}{18}
\][/tex]
#### Conclusion:
Beth's manipulation and step choices are incorrect. She incorrectly applies exponent rules and arrives at a wrong solution [tex]\( x = \frac{1}{18} \)[/tex]. This is not correct.
### 3. Adam's Approach
#### Steps:
1. Adam begins with the same equation:
[tex]\[
25^{2x} = 125^{1/3}
\][/tex]
2. Adam simplifies bases by raising both sides to the power of 3 for easier manipulation:
[tex]\[
(25^{2x})^3 = (125^{1/3})^3
\][/tex]
3. This yields:
[tex]\[
25^{6x} = 125
\][/tex]
4. Converting the bases to powers of 5:
[tex]\[
(5^2)^{6x} = 5^3
\][/tex]
5. Applying exponent rules [tex]\((a^m)^n = a^{mn}\)[/tex], we get:
[tex]\[
5^{12x} = 5^3
\][/tex]
6. Equating the exponents:
[tex]\[
12x = 3
\][/tex]
7. Solving for [tex]\(x\)[/tex]:
[tex]\[
x = \frac{3}{12} = \frac{1}{4}
\][/tex]
#### Conclusion:
Adam's steps are somewhat more complex but logically correct. He arrives at the correct solution [tex]\( x = \frac{1}{4} \)[/tex].
### Final Summary:
- Jerry and Adam both solved the problem correctly and arrived at the right answer [tex]\( x = \frac{1}{4} \)[/tex].
- Beth made significant errors in her approach and as a result, her solution [tex]\( x = \frac{1}{18} \)[/tex] is incorrect.
