Megan:
x to the one third power = [tex]x ^{1/3} [/tex]
x to the one twelfth power = [tex]x ^{1/12} [/tex]
The quantity of x to the one third power, over x to the one twelfth power is:
[tex] \frac{x ^{1/3}}{x ^{1/12}} [/tex]
Since [tex] \frac{ x^{a} }{ x^{b} } = x ^{a-b} [/tex]
then [tex]\frac{x ^{1/3}}{x ^{1/12}} = x^{1/3-1/12} [/tex]
Now, just subtract exponents:
1/3 - 1/12 = 4/12 - 1/12 = 3/12 = 1/4
[tex]\frac{x ^{1/3}}{x ^{1/12}} = x^{1/3-1/12} = x^{1/4} [/tex]
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
[tex] \sqrt[32]{x* x^{2} * x^{5} } [/tex]
Since [tex] x^{a}* x^{b}= x^{a+b} [/tex]
Then [tex]\sqrt[32]{x* x^{2} * x^{5} }= \sqrt[32]{ x^{1+2+5} } =\sqrt[32]{ x^{8} }[/tex]
Since [tex] \sqrt[n]{x^{m}} = x^{m/n} } [/tex]
Then [tex]\sqrt[32]{ x^{8} }= x^{8/32} = x^{1/4} [/tex]
Since both Megan and Julie got the same result, it can be concluded that their expressions are equivalent.
