Question

How do you simplify `(\frac { x ^ { 2} } { x ^ { 6} } ) ^ { - \frac { 1} { 2} }`?

Answer 1

`x^2`

Explanation:

Distribute the `-1/2` exponent to each term:

`(x^(2*-1/2))/(x^(6*-1/2))=x^(-1)/x^(-3)`

Rewrite using positive exponents:

`x^3/x`
Note: this is because of negative exponent rule: `x^(-a)=1/x^a` so essentially, `x^(-1)/x^(-3)=(1/x^1)/(1/x^3)=1/x^1*x^3/1=x^3/x^1 or x^3/x`, Also note that `x^1` is the same as `x`

Finally, simplify:
`x^2`

Note: We get this result because of the quotient rule of exponents: `x^a/x^b=x^(a-b)` so essentially, `x^3/x=x^(3-1)=x^2`