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Answer :
To solve the equation [tex]\(3125 = (n+10)^{\frac{5}{3}}\)[/tex], we need to find the value of [tex]\(n\)[/tex].
1. Rearrange the equation: Start with the given equation:
[tex]\((n+10)^{\frac{5}{3}} = 3125\)[/tex]
2. Isolate [tex]\((n+10)\)[/tex] by raising both sides to the reciprocal power:
To solve for [tex]\((n+10)\)[/tex], raise both sides of the equation to the power of [tex]\(\frac{3}{5}\)[/tex].
This means we compute:
[tex]\[
n+10 = 3125^{\frac{3}{5}}
\][/tex]
3. Calculate [tex]\(3125^{\frac{3}{5}}\)[/tex]:
The expression [tex]\(3125^{\frac{3}{5}}\)[/tex] represents taking the fifth root of 3125 and then cubing the result.
- The fifth root of 3125 is 5, because [tex]\(5^5 = 3125\)[/tex].
- Cubing 5 gives: [tex]\(5^3 = 125\)[/tex].
Therefore, [tex]\(3125^{\frac{3}{5}} = 125\)[/tex].
4. Solve for [tex]\(n\)[/tex]:
Now that we know [tex]\(n+10 = 125\)[/tex], we subtract 10 from both sides to solve for [tex]\(n\)[/tex]:
[tex]\[
n = 125 - 10 = 115
\][/tex]
Thus, the value of [tex]\(n\)[/tex] is 115.
1. Rearrange the equation: Start with the given equation:
[tex]\((n+10)^{\frac{5}{3}} = 3125\)[/tex]
2. Isolate [tex]\((n+10)\)[/tex] by raising both sides to the reciprocal power:
To solve for [tex]\((n+10)\)[/tex], raise both sides of the equation to the power of [tex]\(\frac{3}{5}\)[/tex].
This means we compute:
[tex]\[
n+10 = 3125^{\frac{3}{5}}
\][/tex]
3. Calculate [tex]\(3125^{\frac{3}{5}}\)[/tex]:
The expression [tex]\(3125^{\frac{3}{5}}\)[/tex] represents taking the fifth root of 3125 and then cubing the result.
- The fifth root of 3125 is 5, because [tex]\(5^5 = 3125\)[/tex].
- Cubing 5 gives: [tex]\(5^3 = 125\)[/tex].
Therefore, [tex]\(3125^{\frac{3}{5}} = 125\)[/tex].
4. Solve for [tex]\(n\)[/tex]:
Now that we know [tex]\(n+10 = 125\)[/tex], we subtract 10 from both sides to solve for [tex]\(n\)[/tex]:
[tex]\[
n = 125 - 10 = 115
\][/tex]
Thus, the value of [tex]\(n\)[/tex] is 115.
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