Sure! Let's go through the step-by-step process to evaluate the expression [tex]\(\left(\frac{R}{R+h}\right)^n\)[/tex] with the given numerical values.
Given:
- [tex]\( R = 10 \)[/tex]
- [tex]\( h = 5 \)[/tex]
- [tex]\( n = 3 \)[/tex]
We need to calculate the expression [tex]\(\left(\frac{R}{R+h}\right)^n\)[/tex].
1. Substitute the values of [tex]\(R\)[/tex] and [tex]\(h\)[/tex] into the expression:
[tex]\[
\frac{R}{R + h} = \frac{10}{10 + 5}
\][/tex]
2. Simplify the denominator inside the fraction:
[tex]\[
10 + 5 = 15
\][/tex]
So, the fraction becomes:
[tex]\[
\frac{10}{15}
\][/tex]
3. Simplify the fraction:
[tex]\[
\frac{10}{15} = \frac{2}{3}
\][/tex]
4. Substitute this simplified fraction back into the expression, raised to the power of [tex]\(n\)[/tex]:
[tex]\[
\left(\frac{2}{3}\right)^3
\][/tex]
5. Calculate the power:
[tex]\[
\left(\frac{2}{3}\right)^3 = \frac{2^3}{3^3} = \frac{8}{27}
\][/tex]
6. Convert the fraction into a decimal (if needed):
[tex]\[
\frac{8}{27} \approx 0.2962962962962962
\][/tex]
Hence, the final result of the expression [tex]\(\left(\frac{R}{R+h}\right)^n\)[/tex] with the given values is approximately [tex]\(0.2962962962962962\)[/tex].
