1. 5.1 Radicals & Rational Exponents
Objectives:
Define and apply rational & radical exponents.
Simplify expressions containing radicals or rational exponents.

2. Simplifying Radicals
Although we have been simplifying radicals such as square roots, the process we use doesn’t work as well for higher roots like cube roots or fourth roots. The first examples start with problems that could be simplified with other methods, but we will introduce the new method which will be applied to more complicated cases.

3. Example #1 Simplify each expression.
First break down all numbers into their prime factorization.
Divide the remaining exponents of the factors by the index.
If it divides evenly the factor can be pulled out however many times it divides evenly, remainders remain under the radical as the exponent of the same factor.
Remember that square roots have an index of 2. You can multiply roots with the same index.
For this problem, the index of 2 divides into the exponent of 4 twice. That answer becomes the exponent of the same factor outside the radical.
The exponent on the 3 is a 1 and does not divide by 2 evenly so it remains under the radical.

4. Example #1 Simplify each expression.
This time the index of 2 divided into the exponent on the factor 3 once so the answer went on the factor of 3 outside the radical. The exponent on the 5 didn’t divide evenly so it remained under the radical.
For the second radical, the index of 2 divided into the exponent of 3 once as well, so 1 factor of 5 was pulled out, but the remainder was also a 1 so it became the exponent on the 5 inside the radical.
After being simplified, the radicands are the same so they can be combined.
Even though they have the same index they cannot be combined because they aren’t being multiplied or divided.

5. Example #1 Simplify each expression.
The process for working with variables is the same as with numbers.
This time the index of 3 went into the exponents on the 2, 5, and x evenly, so 1 of each factor was pulled out.
The index divided into the exponent on the y twice, so 2 became the exponent on the y outside the radical, but a remainder of 1 was left and became the exponent on the y inside the radical.

6. Example #1 Simplify each expression.
Although FOIL could be used, these are conjugates of each other and the middle terms would cancel.
The b > 0 is important because it tells us we don’t have to worry about imaginary values.

7. Example #1 Simplify each expression.

8. Example #1 Simplify each expression.
The radicals in the bottom are combined since the index is the same. Additionally, like factors of 3 in the numerator and denominator can cancel for the same reason. That is where the 32 comes from up top.
Although a square root of a negative number is imaginary, the cube root of −1 is −1.

9. Example #1 Simplify each expression.
This problem can be simplified using a slight number trick. Rewrite this number as a product of 81 and a power of 10 similar to scientific notation.
Be careful when simplifying with negative exponents. If a remainder remains, a negative goes on the exponent inside AND outside the radical.
(Note: This didn’t apply to this problem)

10. Properties of Radical & Rational Exponent Expressions

11. Laws of Exponents

12. Example #2 Rewrite as a radical & simplify
Example #2 Rewrite as a radical & simplify. Verify your answer on a calculator.

13. Example #2 Rewrite as a radical & simplify
Example #2 Rewrite as a radical & simplify. Verify your answer on a calculator.
Never put the negative from the exponent in the index.

14. Example #3 Simplify each expression using only positive exponents.
Negative exponents need rewritten as positive by moving them to the denominator.
The c1 in the denominator got moved to the numerator and became c−1, then combined with c−4 to become c−5. Notice when simplified, the negative exponent went inside AND outside the radical.

15. Example #4 Simplify each expression using only positive exponents.
Rewrite numbers into their prime factorization. Power to a power rule says to multiply exponents.
Rewrite radicals as fraction exponents. When multiplying like bases, ADD the exponents.

16. Example #5 Simplify each expression using only positive exponents.
Distribute. Remember when multiplying like bases, ADD the exponents.

17. Example #5 Simplify each expression using only positive exponents.
First groups with negative exponents are rearranged. Groups with the same base multiplied together can be combined by adding the exponents as well.
The final answer has all like bases combined with no negative exponents.

18. Example #6 Write the expression without radicals using only positive exponents.
Remember with power to a power to multiply exponents. Add exponents for bases that are the same.

19. Example #6 Write the expression without radicals using only positive exponents.
Here we have two radicals buried inside each other. The same rules apply, they become fractional exponents and with power to a power you multiply the exponents together.

20. Example #6 Write the expression without radicals using only positive exponents.

21. Example #7 Rationalize the denominator of each fraction
Example #7 Rationalize the denominator of each fraction. (Remove the radical from the denominator)
Multiply the top and bottom by the conjugate of the denominator.

22. Example #8 Factor each expression.
Factoring expressions like this works very similarly to normal factoring.
Note: The exponent on each x term always matches that of the middle term.
Here the middle term is missing so this is a difference of squares and they must add to be 0.
This expression also factors twice.

23. Example #9 Assume h ≠ 0, rationalize the numerator.
Multiply the top and bottom by the conjugate of the numerator this time.