This document discusses solving various types of radical and rational exponent equations. It provides steps for solving square root, other radical, and rational exponent equations. Examples are worked through for each type. The document emphasizes that solutions must always be checked for extraneous solutions, as deriving false solutions is possible when solving these types of equations. Homework practice problems are assigned from the textbook.

1. 6.5 Solving Square Root and
Other Radical Equations

2. Radical equations
• A radical equation is an equation that
has a variable in a radicand or a variable
with a rational exponent.
– If the radical has index 2, then the equation is
a square root equation.

3. Solving Radical Equations
1. Isolate the radical
2. Raise each side to the power suggested
by the index
3. Simplify
4. Check

4. Example: Solve
3 + 2x − 3 = 8

5. Example: Solve
4x + 1 − 5 = 0

6. Solving Equations with Rational
Exponents
1. Isolate the quantity that has the rational
exponent
2. Raise each side of the equation to the
reciprocal power
3. Simplify
4. Check
Note: If either m or n is even then

7. Example: Solve
2
3 ( x + 1) = 12
3

8. Example: Solve
2
2 ( x + 3) = 8
3

9. Example: Solve
( x + 1)
3
3 5
+ 1 = 25

10. Extraneous Solutions
• When we are solving radical equations, it
is possible to derive extraneous solutions,
therefore we must ALWAYS check our
solutions.
– Remember: an extraneous solution gives a
false statement when checked

11. Example: Solve. Check for
extraneous solutions
x+7 −5= x

12. Example: Solve. Check for
extraneous solutions
5x − 1 + 3 = x

13. Solving an equation with two radical
expressions (or two terms with rational
exponents)
• Isolate one of the radical (or one of the terms
with a rational exponent) (Hint: Choose the
more complicated one first)
• Eliminate the radical (or the rational exponent)
• Simplify
• Isolate the other radical (or term with the
rational exponent)
• Eliminate the radical (or the rational exponent)
• Simplify
• Check for extraneous solutions

14. Example: Solve. Check for
extraneous solutions
2x + 1 − x = 1

15. Example: Solve. Check for
extraneous solutions
5x + 4 − x = 4

16. Example: Solve. Check for
extraneous solutions
1 1
( x − 2) 2 − ( 28 − 2 x ) = 0
4

17. Homework
• P395 #9, 13, 17, 19, 21, 23, 25 – 44 every
other odd, 49, 53, 55, 56