https://file.aiquickdraw.com/mathbot/file/1726698645854n2jnyplk.jpg

Let's solve the two math problems visible in the image.

### Problem 5:
\[
\left(a^8\right)^{\frac{3}{2}}
\]
To solve this, we use the property of exponents:
\[
\left(a^m\right)^n = a^{m \cdot n}
\]
Applying this to the expression:
\[
\left(a^8\right)^{\frac{3}{2}} = a^{8 \cdot \frac{3}{2}} = a^{12}
\]
Thus, the simplified form of \(\left(a^8\right)^{\frac{3}{2}}\) is:
\[
a^{12}
\]

---

### Problem 7:
\[
\left(81x^{12}\right)^{1.25}
\]
First, express \(1.25\) as a fraction:
\[
1.25 = \frac{5}{4}
\]
Now, apply the exponent to both parts inside the parentheses. We break the expression into two parts: \(81^{\frac{5}{4}}\) and \(\left(x^{12}\right)^{\frac{5}{4}}\).

- For \(81^{\frac{5}{4}}\), recognize that \(81 = 3^4\), so:
\[
81^{\frac{5}{4}} = (3^4)^{\frac{5}{4}} = 3^{4 \cdot \frac{5}{4}} = 3^5 = 243
\]

- For \(\left(x^{12}\right)^{\frac{5}{4}}\), use the property of exponents:
\[
(x^{12})^{\frac{5}{4}} = x^{12 \cdot \frac{5}{4}} = x^{15}
\]

So, the final expression is:
\[
243x^{15}
\]

---

### Final Answers:
1. \(\left(a^8\right)^{\frac{3}{2}} = a^{12}\)
2. \(\left(81x^{12}\right)^{1.25} = 243x^{15}\)

Would you like further details or clarification?

Here are 5 related questions to consider:
1. What is the general rule for multiplying exponents?
2. How would the result change in problem 5 if the exponent was \(\frac{5}{3}\)?
3. How do you solve expressions with negative exponents?
4. What happens when a variable in the base has an exponent of zero?
5. How do fractional exponents relate to radicals?

**Tip**: When raising a power to another power, always multiply the exponents!

Simplify (a^8)^(3/2) and (81x^12)^(1.25).

Learn how to simplify the expressions (a^8)^(3/2) and (81x^12)^(1.25) using the laws of exponents. This problem demonstrates exponent multiplication and fractional exponents. Suitable for Grades 9-11.

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