Question

Need help please help me

Megan and Julie are stuck simplifying radical expressions. Megan has to simplify the quantity of x to the one third power, over x to the one twelfth power. Julie has to simplify the thirty second root of the quantity of x times x to the second times x to the fifth. Using full sentences describe how to fully simplify Megan and Julie’s expressions. Describe if Megan and Julie started with equivalent expressions or if they started with expressions that are not equal.

Answer 1

Megan:
x to the one third power =
`x ^(1/3)`
x to the one twelfth power =
`x ^(1/12)`

The quantity of x to the one third power, over x to the one twelfth power is:

`(x ^(1/3))/(x ^(1/12))`

Since
`( x^(a) )/( x^(b) ) = x ^(a-b)`
then
`(x ^(1/3))/(x ^(1/12)) = x^(1/3-1/12)`

Now, just subtract exponents:
1/3 - 1/12 = 4/12 - 1/12 = 3/12 = 1/4


`(x ^(1/3))/(x ^(1/12)) = x^(1/3-1/12) = x^(1/4)`


Julie:
x times x to the second times x to the fifth = x * x² * x⁵

The thirty second root of the quantity of x times x to the second times x to the fifth is

`\sqrt[32]{x* x^(2) * x^(5) }`

Since
`x^(a)* x^(b)= x^(a+b)`
Then
`\sqrt[32]{x* x^(2) * x^(5) }= \sqrt[32]{ x^(1+2+5) } =\sqrt[32]{ x^(8) }`

Since
`\sqrt[n]{x^(m)} = x^(m/n) }`
Then
`\sqrt[32]{ x^(8) }= x^(8/32) = x^(1/4)`

Since both Megan and Julie got the same result, it can be concluded that their expressions are equivalent.