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Simplifying Expressions with Rational Exponents

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  • 0:05 Expressions with…
  • 0:41 Examples 1-4
  • 2:19 Examples 5-7
  • 6:07 Radical to Rational…
  • 6:27 Example 8
  • 7:17 Lesson Summary
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Lesson Transcript
Instructor: Kathryn Maloney

Kathryn teaches college math. She holds a master's degree in Learning and Technology.

Simplifying expressions with rational exponents is so easy. In fact, you already know how to do it! We simply use the exponent properties but with fractions as the exponent!

Expressions with Rational Exponents

Rational exponents follow exponent properties except using fractions.

Review of exponent properties - you need to memorize these. Just can't seem to memorize them? Have you tried flashcards? They work fantastic, and you can even use them anywhere!

  1. Product of Powers: xa*xb = x(a + b)
  2. Power to a Power: (xa)b = x(a * b)
  3. Quotient of Powers: (xa)/(xb) = x(a - b)
  4. Power of a Product: (xy)a = xaya
  5. Power of a Quotient: (x/y)a = xa / ya
  6. Negative Exponent: x(-a) = 1 / xa
  7. Zero Exponent: x0 = 1

Putting the exponent rules to work with exponent properties...

Example #1

y(1/2) * y(1/3)

For this one, we're going to follow the product of powers. Remember, when we multiply, we add their exponents.

1/2 + 1/3 = 5/6

So the answer is going to be y(5/6).

Example 2 uses the quotient of powers property to find the solution
Quotient of Powers Example Problem

Example #2

Simplify: x(3/5) / x(2/3)

For this one, we're going to use the quotient of powers. Remember, when we divide, we subtract their exponents. So, we're going to have:

x(3/5 - 2/3)

3/5 - 2/3 = -1/15

So our answer is x(-1/15).

Example #3

Simplify: x(-2/7)

For this one, we're going to use the negative exponents property. Remember, when we have a negative exponent, we flip it. If it's in the numerator, we flip it to the denominator, which is in this case.

So our answer is going to be 1 / (x(2/7)).

Example #4

Simplify: (x(4/5))(3/4)

In this one, we have power to a power. We're going to have (x(4/5))(3/4), so we're going to multiply 4/5 * 3/4 which is 12/20. We need to reduce our fractions when we're going to get our final answer. 12/20 reduces to 3/5.

So our answer is going to be x(3/5).

After reducing 12/20 in example 4, the final answer is x^(3/5)
Power to a Power Example Problem

Example #5

Putting multiple exponent rules to work with exponent properties... Simplify using positive exponents. Always reduce the fractions to lowest terms.


Example 5


First we're going to simplify the power to a power. So now we'll have:


Example 5.1


Write like terms over each other, if necessary. Well, we already have the p's over the p's and the q's over the q's. There's no need to simplify fractions now. We're going to go right to simplifying quotient of powers. Remember, when we divide, we subtract their exponents. So we have:

p(2/6 - 1/2) * q(6/3 - 1/2)

That gives us:

p(-1/6) * q(9/6)

Next we need to reduce the fractions because we're almost to our answer. So we'll have:

p(-1/6) * q(3/2)

We want to rewrite these using positive exponents. Remember, if it's negative in the numerator, it flips to the denominator. So our final answer's going to be:

q(3/2) / p(1/6)

Example #6

The solution for example 6 after applying the quotient of powers property
Quotient of Powers Example 6 Solution

Simplify using positive exponents. Always reduce the fractions to lowest terms. We're going to have:


Example6


We're going to simplify power to a power. So we'll have:


example6.1


Remember, power to a power means to multiply the exponents. Next, let's write like terms over each other. We already have 23 over 82 and m(6/3) over m(2/6). So let's move to the next step. There's no need to simplify fractions just yet, so we're going to simplify quotient of powers. Remember, when we divide, we subtract. So now we're going to have:

8/64 * m(6/3 - 2/6)

Well, 8/64 is 1/8. m to the 6/3 - 2/6 is m to the 10/6. So it turns out that our final answer is:

m(5/3) / 8

We won't touch the improper fraction in this video. We're just simplifying rational exponents.

Example #7

Simplify using positive exponents. Always reduce fractions to lowest terms.


example7


First we're going to simplify power to a power. Remember, power to a power means to multiply the exponents. That'll give us:


example7.1


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