1. Working with Fractional Powers

Not sure how to solve the last part of this problem. The original equation reads:

$\displaystyle \sqrt{xy^3} (\sqrt{x^5y} - \sqrt{xy^7})$

So then you raise the powers to get some nicer exponents to work which looks like this

$\displaystyle x^\frac{1}{2}y^\frac{3}{2}(x^\frac{5}{2}y^\frac{1} {2}) - x^\frac{1}{2}y^\frac{3}{2}(x^\frac{1}{2}y^\frac{7} {2})$

and here is where I am lost, the author adds the exponents of the variables and ends up with

$\displaystyle x^\frac{6}{2}y^\frac{4}{2}-x^\frac{2}{2}y^\frac{10}{2}$

How is the author getting x^6/2, y^4/2 when you distribute for the first part of the equation? I am completely clueless as to how she did this could someone please fill me in. Thanks!

2. Originally Posted by cmf0106
Not sure how to solve the last part of this problem. The original equation reads:

$\displaystyle \sqrt{xy^3} (\sqrt{x^5y} - \sqrt{xy^7})$

So then you raise the powers to get some nicer exponents to work which looks like this

$\displaystyle x^\frac{1}{2}y^\frac{3}{2}(x^\frac{5}{2}y^\frac{1} {2}) - x^\frac{1}{2}y^\frac{3}{2}(x^\frac{1}{2}y^\frac{7} {2})$

and here is where I am lost, the author adds the exponents of the variables and ends up with

$\displaystyle x^\frac{6}{2}y^\frac{4}{2}-x^\frac{2}{2}y^\frac{10}{2}$

How is the author getting x^6/2, y^4/2 when you distribute for the first part of the equation? I am completely clueless as to how she did this could someone please fill me in. Thanks!
The 1st term looks like this:

$\displaystyle x^\frac{1}{2}y^\frac{3}{2}(x^\frac{5}{2}y^\frac{1} {2})$

To find the new exponent of x, add the two different exponents because this property holds:

$\displaystyle x^a * x^b = x^{a+b}$

Therefore: $\displaystyle \frac{1}{2}+\frac{5}{2}=3$, and the new exponent for x is 3. Likewise, y now has an exponent of 2.

3. Hello,

Originally Posted by cmf0106
Not sure how to solve the last part of this problem. The original equation reads:

$\displaystyle \sqrt{xy^3} (\sqrt{x^5y} - \sqrt{xy^7})$

So then you raise the powers to get some nicer exponents to work which looks like this

$\displaystyle x^\frac{1}{2}y^\frac{3}{2}(x^\frac{5}{2}y^\frac{1} {2}) - x^\frac{1}{2}y^\frac{3}{2}(x^\frac{1}{2}y^\frac{7} {2})$

and here is where I am lost, the author adds the exponents of the variables and ends up with

$\displaystyle x^\frac{6}{2}y^\frac{4}{2}-x^\frac{2}{2}y^\frac{10}{2}$

How is the author getting x^6/2, y^4/2 when you distribute for the first part of the equation? I am completely clueless as to how she did this could someone please fill me in. Thanks!
The rule states : $\displaystyle a^b a^c=a^{b+c}$

$\displaystyle x^{\frac 12}y^{\frac 32}(x^{\frac 52}y^{\frac 12})=x^{\frac 12}y^{\frac 32}x^{\frac 52}y^{\frac 12}=x^{\frac 12}x^{\frac 52}y^{\frac 32}y^{\frac 12}=\dots$

4. Thanks for the input everyone! I just realized I made a mistake. At times I forgot the author assumes that one is not utilizing a calculator.