To solve the equation [tex]\(\sqrt[3]{5^{x+2}} = 25\)[/tex], let's go through the steps methodically.
1. Rewrite the equation in exponential form:
[tex]\[\sqrt[3]{5^{x+2}} = (5^{x+2})^{1/3}\][/tex]
The equation then becomes:
[tex]\[(5^{x+2})^{1/3} = 25\][/tex]
2. Eliminate the cube root by raising both sides to the power of 3:
[tex]\[((5^{x+2})^{1/3})^3 = 25^3\][/tex]
Simplifying the left side, we get:
[tex]\[5^{x+2} = 25^3\][/tex]
3. Express 25 in terms of the base 5:
Recall that [tex]\(25 = 5^2\)[/tex].
Therefore:
[tex]\[25^3 = (5^2)^3\][/tex]
4. Simplify the right side using the properties of exponents:
[tex]\[(5^2)^3 = 5^{2 \cdot 3} = 5^6\][/tex]
Now the equation is:
[tex]\[5^{x+2} = 5^6\][/tex]
5. Since the bases are the same, set the exponents equal to each other:
[tex]\[x + 2 = 6\][/tex]
6. Solve for [tex]\(x\)[/tex]:
[tex]\[x = 6 - 2\][/tex]
[tex]\[x = 4\][/tex]
Thus, the solution to the equation [tex]\(\sqrt[3]{5^{x+2}} = 25\)[/tex] is:
[tex]\[
x = 4
\][/tex]
