Question

See #1 for an example. Leave your answer in exponent form.

(1 point)

8^9/8^4=8^5

x^9/x=x^8

8/8^2/3=8^1/3

5^2/^5√25=

^2√11^3/^5√11^4

^3√216/27^1/3

Answer 1

Let's simplify each expression step by step, leaving the final answer in exponent form.

1. **\( \frac{8^9}{8^4} \)**
Using the property of exponents \( \frac{a^m}{a^n} = a^{m-n} \):
\[
\frac{8^9}{8^4} = 8^{9-4} = 8^5
\]

2. **\( \frac{x^9}{x} \)**
Again applying \( \frac{a^m}{a^n} = a^{m-n} \):
\[
\frac{x^9}{x} = x^{9-1} = x^8
\]

3. **\( \frac{8}{8^{2/3}} \)**
Rewriting 8 as \( 8^1 \):
\[
\frac{8^1}{8^{2/3}} = 8^{1 - 2/3} = 8^{3/3 - 2/3} = 8^{1/3}
\]

4. **\( 5^{2/\sqrt[5]{25}} \)**
We know \( 25 = 5^2 \), so \( \sqrt[5]{25} = \sqrt[5]{5^2} = 5^{2/5} \):
\[
5^{2/\sqrt[5]{25}} = 5^{2/(5^{2/5})} = 5^{2 \cdot 5^{-2/5}} = 5^{2-2/5} = 5^{10/5 - 2/5} = 5^{8/5}
\]

5. **\( \frac{\sqrt{11^3}}{\sqrt[5]{11^4}} \)**
Rewriting the square root and the fifth root:
\[
\frac{\sqrt{11^3}}{\sqrt[5]{11^4}} = \frac{11^{3/2}}{11^{4/5}} = 11^{3/2 - 4/5}
\]
To combine the exponents, find a common denominator (10):
\[
3/2 = 15/10 \quad \text{and} \quad 4/5 = 8/10 \quad \Rightarrow \quad 3/2 - 4/5 = 15/10 - 8/10 = 7/10
\]
So we have:
\[
\frac{\sqrt{11^3}}{\sqrt[5]{11^4}} = 11^{7/10}
\]

6. **\( \frac{\sqrt[3]{216}}{27^{1/3}} \)**
We know that \( 216 = 6^3 \) and \( 27 = 3^3 \):
\[
\frac{\sqrt[3]{216}}{27^{1/3}} = \frac{216^{1/3}}{27^{1/3}} = \frac{(6^3)^{1/3}}{(3^3)^{1/3}} = \frac{6^{3/3}}{3^{3/3}} = \frac{6^1}{3^1} = \frac{6}{3} = 2
\]
In exponent form, we can express this as \( 2^1 \).

### Final Results:
1. \( 8^5 \)
2. \( x^8 \)
3. \( 8^{1/3} \)
4. \( 5^{8/5} \)
5. \( 11^{7/10} \)
6. \( 2^1 \)