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Solve for X: 25^x = 125, Find X^2
Published on September 25, 2024
Algebra
Solve the math problem: 25^x = 125. Find the value of x^2. The answer is 9/4 or 2.25.
Question
Answer
Let’s break down this problem step-by-step.
1. Understand the First Equation
- We have 25<sup>x</sup> = 125. This means we need to find what power (x) we raise 25 to in order to get 125.
2. Simplify by Finding a Common Base
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Notice that both 25 and 125 can be expressed as powers of 5:
- 25 = 5<sup>2</sup>
- 125 = 5<sup>3</sup>
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Substitute these into the first equation: (5<sup>2</sup>)<sup>x</sup> = 5<sup>3</sup>
3. Solve for x
- Using the rule of exponents that says (a<sup>m</sup>)<sup>n</sup> = a<sup>m*n</sup>, we get: 5<sup>2x</sup> = 5<sup>3</sup>
- Since the bases are now the same, we can equate the exponents: 2x = 3
- Solve for x by dividing both sides by 2: x = 3/2
4. Calculate x²
- Now that we know x = 3/2, we can find x²:
- x² = (3/2)² = (3/2) * (3/2) = 9/4
Answer:
Therefore, x² = 9/4 or 2.25