Question

Simplify the expression.

x^2/3 y^-1/4 over x^1/2 y^-1/2

Answer 1

\[\frac{x^{2/3}y^{-1/4}}{x^{1/2}y^{-1/2}}\]

Answer 2

Yes, that. I tried posting that but apparently it didn't go through. Can you explain how to do it?

Answer 3

@Hero

Answer 4

Hang on

Answer 5

Basically, in general negative exponents do not mean negative number. They mean "inverse" as in "fractional reciprocal". In other words something like:

\(y^{-x}\) "means the the multiplicative inverse of \(y^x\)" or simply \(\dfrac{1}{y^x}\)

So in summary, \(y^{-x} = \dfrac{1}{y^x}\)

Answer 6

Similarly \(\dfrac{1}{y^{-x}} = y^x\)

Answer 7

In other words, if you see a negative fractional exponent in the numerator, you can move the expression to the denominator and remove the negative and it will be an equivalent form.

Answer 8

Thus:

\(\dfrac{x^{2/3}y^{-1/4}}{x^{1/2}y^{-1/2}} = \dfrac{x^{2/3}y^{1/2}}{x^{1/2}y^{1/4}}\)

Answer 9

I just reverse the y?

Answer 10

At this point, what you want to do is use this rule of exponents:

If b > c, then \(\dfrac{a^b}{a^c} = a^{b - c}\)
or

If b < c, then \(\dfrac{a^b}{a^c} = \dfrac{1}{a^{c - b}}\)

Answer 11

The reason for this is because you want to avoid negative exponents at all costs during simplification. So always subtract the smaller exponent from the bigger exponent.

Answer 12

In this case, 2/3 > 1/2 and 1/2 > 1/4 therefore, by rule of exponents, we can simplify the expression to:

\(\dfrac{x^{2/3}y^{-1/4}}{x^{1/2}y^{-1/2}} = \dfrac{x^{2/3}y^{1/2}}{x^{1/2}y^{1/4}} = x^{(2/3 - 1/2)}y^{(1/2 - 1/4)}\)

Which, at this point, simplifying becomes a matter of simply subtracting fractions.

Answer 13

And that would be \[x ^{1/6}y ^{1/4}\]

Answer 14

correct?