To solve the equation [tex]\(\sqrt[10]{(x + 10)^2} = \sqrt[5]{2x - 5}\)[/tex], let’s break it down step by step:
1. Rewrite the equation using exponents:
[tex]\[
(x + 10)^{\frac{1}{5}} = (2x - 5)^{\frac{1}{5}}
\][/tex]
2. Remove the fractional exponents by raising both sides to the power of 5:
[tex]\[
x + 10 = (2x - 5)^{2}
\][/tex]
3. Expand the right-hand side:
[tex]\[
x + 10 = 4x^2 - 20x + 25
\][/tex]
4. Rearrange the equation to set it to zero:
[tex]\[
4x^2 - 20x + 25 - x - 10 = 0
\][/tex]
[tex]\[
4x^2 - 21x + 15 = 0
\][/tex]
5. Now we have a quadratic equation:
[tex]\[
4x^2 - 21x + 15 = 0
\][/tex]
6. Use the quadratic formula to solve for [tex]\(x\)[/tex]:
[tex]\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\][/tex]
Where [tex]\(a = 4\)[/tex], [tex]\(b = -21\)[/tex], and [tex]\(c = 15\)[/tex].
7. Calculate the discriminant:
[tex]\[
\Delta = b^2 - 4ac = (-21)^2 - 4 \cdot 4 \cdot 15 = 441 - 240 = 201
\][/tex]
8. Find the roots:
[tex]\[
x_1 = \frac{-(-21) + \sqrt{201}}{2 \cdot 4} = \frac{21 + \sqrt{201}}{8} \approx 4.397
\][/tex]
[tex]\[
x_2 = \frac{-(-21) - \sqrt{201}}{2 \cdot 4} = \frac{21 - \sqrt{201}}{8} \approx 0.853
\][/tex]
9. Check which of these roots is an answer that matches the provided choices:
Comparing the roots ([tex]\(4.397\)[/tex] and [tex]\(0.853\)[/tex]) with the given options (5, 2, 10, and 15), the closest match is 5.
Therefore, the correct answer is:
[tex]\[
\boxed{5}
\][/tex]
