To simplify the given expression [tex]\( \left(\frac{(x_1^2)^3}{5r^3}\right)^2 \)[/tex] using only positive exponents, we need to follow a step-by-step process:
1. Evaluate the Power Inside the Parentheses:
First, look inside the parentheses. You have [tex]\( (x_1^2)^3 \)[/tex], which is a power raised to another power. When you raise a power to another power, you multiply the exponents:
[tex]\[
(x_1^2)^3 = x_1^{2 \cdot 3} = x_1^6
\][/tex]
2. Simplify the Expression Inside the Parentheses:
Now, substitute [tex]\( x_1^6 \)[/tex] back into the fraction:
[tex]\[
\frac{x_1^6}{5r^3}
\][/tex]
3. Raise the Entire Fraction to the Power of 2:
The entire fraction [tex]\( \frac{x_1^6}{5r^3} \)[/tex] is raised to the power of 2. When you raise a fraction to a power, both the numerator and the denominator are raised to that power:
[tex]\[
\left(\frac{x_1^6}{5r^3}\right)^2 = \frac{(x_1^6)^2}{(5r^3)^2}
\][/tex]
4. Evaluate the Powers Separately:
- For the numerator:
[tex]\[
(x_1^6)^2 = x_1^{6 \cdot 2} = x_1^{12}
\][/tex]
- For the denominator:
[tex]\[
(5r^3)^2 = 5^2 \cdot (r^3)^2 = 25 \cdot r^{3 \cdot 2} = 25r^6
\][/tex]
5. Combine the Results:
Finally, substitute these back into the fraction:
[tex]\[
\left(\frac{(x_1^2)^3}{5r^3}\right)^2 = \frac{x_1^{12}}{25r^6}
\][/tex]
So, the simplified expression, using only positive exponents, is:
[tex]\[
\boxed{\frac{x_1^{12}}{25r^6}}
\][/tex]
