Question

Use the rules of exponents to simplify the expressions. Match the expression with its equivalent value.

(-2)
1.
1-21-10
32
2. 21.24
-32
3.
32
1
4.
2-

Answer 1

1)
`((-2)^(-5))/((-2)^(-10))=-32`

2)
`2^(-1).2^(-4) = (1)/(32)`

3)
`(-(1)/(2) )^3.(-(1)/(2) )^2=-(1)/(32)`

4)
`(2)/(2^(-4)) = 32`

Explanation:

1)
`((-2)^(-5))/((-2)^(-10))`

Solving using exponent rule:
`a^(-m)=(1)/(a^m)`

`((-2)^(-5))/((-2)^(-10))\\=(-2)^(-5+10)\\=(-2)^(5)\\=-32`

So,
`((-2)^(-5))/((-2)^(-10))=-32`

2)
`2^(-1).2^(-4)`

Using the exponent rule:
`a^m.a^n=a^(m+n)`

We have:

`2^(-1).2^(-4)\\=2^(-1-4)\\=2^(-5)`

We also know that:
`a^(-m)=(1)/(a^m)`

Using this rule:

`2^(-5)\\=(1)/(2^5)\\=(1)/(32)`

So,
`2^(-1).2^(-4) = (1)/(32)`

3)
`(-(1)/(2) )^3.(-(1)/(2) )^2`

Solving:

`(-(1)/(2) )^3.(-(1)/(2) )^2\\=(-(1)/(8) ).((1)/(4) )\\=-(1)/(32)`

So,
`(-(1)/(2) )^3.(-(1)/(2) )^2=-(1)/(32)`

4)
`(2)/(2^(-4))`

We know that:
`a^(-m)=(1)/(a^m)`

`(2)/(2^(-4))\\=2* 2^4\\=2(16)\\=32`

So,
`(2)/(2^(-4)) = 32`