Question

Can somebody please match up these problems for me? I provided an image to help.

Answer 1

For this we have the following properties of powers:

`(a ^ m) ^ n = a ^ {m * n}\\\frac {a ^ m} {a ^ n} = a ^ {m-n}\\a ^ m * a ^ n = a ^ {m + n}`

So, we have:

`(a ^ 2) ^ 5 = a ^ {2 * 5} = a ^ {10}\\\frac {a ^ 10} {a ^ 10} = a ^ {10-10} = a ^ 0 = 1\\a ^ 1 * a^(10) = a^(1 + 10) = a^(11)`

`\frac {a ^ {10}} {(a ^ 3) ^ 3} = \frac {a ^ {10}} {a ^ 9} = a ^ {10-9} = a ^ 1 = a\\a ^ 7 * (a ^ 3) ^ 0 = a ^ 7 * a ^ 0 = a ^ 7 * 1 = a ^ 7\\\frac {a ^ 4} {a ^ 3} * a ^ 2 = a ^ {4-3} * a ^ 2 = a ^ 1 * a ^ 2 = a ^ 3`

`(a ^ 1) ^ 5 * \frac {a ^ 5} {a ^ 5} = a ^ 5 * 1 = a ^ 5\\\frac {(a ^ {10}) ^ 2} {a ^ 3 * a ^ 2} = \frac {a ^ {20}} {a ^ 5} = a ^ {20-5} = a ^ {15}\\(a ^ 3) ^ 2 * a ^ 2 = a ^ 6 * a ^ 2 = a ^ 8`

`\frac {(a ^ 2) ^ 6} {(a ^ 5) ^ 2} = \frac {a ^ {12}} {a ^ {10}} = a ^ 2`

Answer:

1 C

2.I

3.G

4.H

5.A

6.J

7.B

8.D

9.F

10.E