Question

Simplify the following expressions. Leave your answers in exponent form; however, make sure there aren’t any negative exponents in your answer.
A.
5
2
×

5
6
5
2
× 5
6
=

58


B.
8
19
8
6
8
6

8
19

​
=



C.
3
2

×

3
−
5
3
2
× 3
−5
=



D.
(
4
3
)
5
(4
3
)
5
=



E.
(
x
3
)
2
(x
3
)
2
=



F.
3
9
3
−
5
3
−5

3
9

​
=



G.
4
3
4
7
4
7

4
3

​
=



H.
(
x
4
)
(
x
6
)
(x
4
)(x
6
) =

Answer 1

Let's simplify each of the given expressions one by one, ensuring that no negative exponents appear in the final answers.

### A.
\[
5^2 \times 5^6 = 5^{2+6} = 5^8
\]

### B.
\[
8^{19} \times 8^{6} = 8^{19+6} = 8^{25}
\]

### C.
\[
3^2 \times 3^{-5} = 3^{2-5} = 3^{-3} = \frac{1}{3^3} \text{ (not in exponent form, converting to positive exponent)}
\]

Since we need positive exponents:
\[
= \frac{1}{27}
\]

### D.
\[
(4^3)^5 = 4^{3 \times 5} = 4^{15}
\]

### E.
\[
(x^3)^2 = x^{3 \times 2} = x^{6}
\]

### F.
\[
3^9 \times 3^{-5} = 3^{9-5} = 3^{4}
\]

### G.
\[
4^3 \times 4^7 = 4^{3+7} = 4^{10}
\]

### H.
\[
(x^4) \times (x^6) = x^{4+6} = x^{10}
\]

### Summary of Answers:
- A. \( 5^8 \)
- B. \( 8^{25} \)
- C. \( \frac{1}{3^3} \) or \( 3^{-3} \)
- D. \( 4^{15} \)
- E. \( x^{6} \)
- F. \( 3^{4} \)
- G. \( 4^{10} \)
- H. \( x^{10} \)