Question

(x+5)1/2=(5-2x)1/4 This equation has two different rational exponents, 1/2 and 1/4 . The inverse powers for these two rational exponents would be 2 and 4, respectively. Will raising both sides to the 2nd power eliminate both rational exponents? Explain.

Answer 1

No. It will eliminate the rational exponent on the left side completely, but raising 1/4 to the 2nd power still leaves a rational exponent of 1/2 on the right side because 1/4 • 2 = 1/2.

Explanation:

Edmentum

Answer 2

Raising both sides to the second power will eliminate only one rational exponent

Explanation:

we have

`(x+5)^{(1)/(2)}=(5-2x)^{(1)/(4)}`

If raising both sides to the 2nd power

`[(x+5)^{(1)/(2)}]^2=[(5-2x)^{(1)/(4)}]^2`

Remember the property of exponents

`(x^(m) )^(n) =x^(m*n)`

`x^{(m)/(n)}=\sqrt[n]{x^m}`

so

Multiply the exponents

`(x+5)^{(2)/(2)}=(5-2x)^{(2)/(4)}`

Simplify

`(x+5)=(5-2x)^(1)/(2)`

`(x+5)=√(5-2x)`

therefore

Raising both sides to the second power will eliminate only one rational exponent

To eliminate both rational exponents, both sides must be raised to the fourth power

so

`[(x+5)^{(1)/(2)}]^4=[(5-2x)^{(1)/(4)}]^4`

`(x+5)^{(4)/(2)}=(5-2x)^{(4)/(4)}`

simplify

`(x+5)^2=(5-2x)`