Rational Exponents - Simplify part 2 - Youtube.mp4

(male narrator) In Part 1 of this video, we reviewed our exponent properties and talked about how we could use those exponent properties to simplify expressions with rational exponents. Here is another problem with rational exponents that also has negative exponents. We may recall with negative exponents, we simply need to move... those negative exponents to the opposite location. As we do this, the x to the 3/2 stays. The y to the -1/3 moves to the denominator as y to the +1/3. In the denominator, we have an x to the 1/4, a y to the 2/3, and the x to the -5/2 must move up to the numerator. Let's continue working inside the parentheses before we worry about the -1/8 outside of the parentheses. We'll want to simplify the numerator by combining the x's into one single x. Again, we might want some chicken scratch to help us do that: 3/2 plus 5/2--already have a common denominator--is 8/2. But 8/2 reduces to 4. In the numerator, we simply have x to the 4th power. In the denominator, there is an x that has no one to combine with it. We'll write x to the 1/4, but we can also combine the y's. As we combine the y's, we want to add the exponents: 1/3 plus 2/3 is 3/3, or 1. We're left with simply y to the 1st power. All of this is still with the -1/8 as the external exponent. We can continue simplifying by now trying to combine the x's. With division, we subtract exponents: 4 minus 1/4. To make the 4 into a fraction, we put it over 1, and to subtract, we need a common denominator of 4. Multiplying the first fraction by 4 over 4 gives us 16/4, minus 1/4, which is 15/4. This is the new exponent on x, and because it's positive, x to the 15/4 is in the numerator, and y is in the denominator. All of this with -1/8 as the external exponent. When the exponent outside of parentheses is negative, it will simply take the reciprocal of that internal fraction. So now, we move the x to the 15/4 down and the y up. Now, the exponent of 1/8 is positive. We can also use the quotient rule to take this 1/8 and put it onto each factor. As we do, we have y to the 1/8, over x, and with these two exponents, we will multiply them: 1/8 times 15, over 4. To multiply fractions, we multiply straight across, giving us 15 over 32. This is our new exponent on the x. By using the same exponent properties we always have, we can simplify some quite complex-looking expressions with rational exponents by keeping in mind order of operations and either adding, subtracting, or multiplying our exponents.