To solve the equation [tex]\((\sqrt[3]{5})^{x-1} = 25\)[/tex], follow these steps:
1. Understand the given equation:
- The equation is [tex]\((\sqrt[3]{5})^{x-1} = 25\)[/tex].
- [tex]\(\sqrt[3]{5}\)[/tex] means the cube root of 5, which is equivalent to [tex]\(5^{1/3}\)[/tex].
2. Rewrite the equation using exponential notation:
- Rewrite the given equation as [tex]\((5^{1/3})^{x-1} = 25\)[/tex].
3. Simplify the left side using the properties of exponents:
- Use the property of exponents [tex]\((a^m)^n = a^{m \cdot n}\)[/tex] to rewrite the left-hand side.
- [tex]\((5^{1/3})^{x-1} = 5^{(1/3) \cdot (x-1)} = 5^{(x-1)/3}\)[/tex].
4. Equate the bases:
- We now have the equation [tex]\(5^{(x-1)/3} = 25\)[/tex].
5. Express 25 as a power of 5:
- Observe that [tex]\(25 = 5^2\)[/tex].
6. Rewrite the equation with the same base:
- Replace 25 with [tex]\(5^2\)[/tex] to get [tex]\(5^{(x-1)/3} = 5^2\)[/tex].
7. Set the exponents equal to each other:
- Since the bases are the same (both are 5), we can set the exponents equal to each other.
- [tex]\((x-1)/3 = 2\)[/tex].
8. Solve for [tex]\(x\)[/tex]:
- To isolate [tex]\(x\)[/tex], multiply both sides of the equation by 3:
[tex]\[
\frac{x-1}{3} = 2 \\
x-1 = 2 \cdot 3 \\
x-1 = 6
\][/tex]
- Add 1 to both sides of the equation to solve for [tex]\(x\)[/tex]:
[tex]\[
x = 6 + 1 \\
x = 7
\][/tex]
So, the solution to the equation [tex]\((\sqrt[3]{5})^{x-1} = 25\)[/tex] is [tex]\(x = 7\)[/tex].
