1. Exponents and Radicals
MAT 205 FALL 2008
Exponents and Radicals

2. This symbol is the radical or the radical sign
Radical Expressions
Finding a root of a number is the inverse operation of raising a number to a power.
This symbol is the radical or the radical sign
radical sign
index
radicand
The expression under the radical sign is the radicand.
The index defines the root to be taken.

3. Fractional (Rational) Exponents
Note: These properties are valid as long as does not involve the even root of a negative number.
power
root

4. Fractions as exponents
Under Radical
(exponent)
Outside Radical
(root)

5. Fractions as exponents
as exponent means
as exponent means
as exponent means
as exponent means

6. An nth root of any number a is a number whose nth power is a.
nth Roots
An nth root of any number a is a number whose nth power is a.
Examples:

7. Examples:

8. Negative Exponents:
Remember that a negative in the exponent does not make the number negative!
If a base has a negative exponent, that indicates it is in the “wrong” position in fraction. That base can be moved across the fraction bar and given a positive exponent.
EXAMPLES:

9. or or
Examples: or

10. In general, a radical expression is simplified when:
The radicand contains no fractions.
No radicals appear in the denominator = (Rationalization)
The radicand contains no factors that are nth powers of an integer or polynomial.

11. Use the properties of exponents to simplify each expression

12. Example

13. Simplifying Radicals
Let a and b represent positive real numbers.

14. Simplifying Radicals
Remember:

15. Simplify Radical expression

16. Simplify.

17. Simplify

18. simplify

19. Product Property of Radicals
Factor into cubes if possible
Product Property of Radicals

20. Simplifying Radicals

21. To reduce a radical to simplest form:
Remove all perfect nth power factors from a radical of order n.
If a fraction appears under the radical or there is a radical in the denominator of the expression, simplify by rationalizing the denominator.
If possible, reduce the order of the radical.

22. Multiplication & Division of Radicals
To multiply expressions containing radicals, we will use the property
where a and b represent positive values.
Notice: that the order (indexes) of the radicals being multiplied must be the same.

24. Rationalizing

25. This cannot be divided which leaves the radical in the denominator
This cannot be divided which leaves the radical in the denominator. We do not leave radicals in the denominator. So we need to rationalize by multiplying the fraction by something so we can eliminate the radical in the denominator.
42 cannot be simplified, so we are finished.

26. Simplify

27. Rationalize the denominator
Simplify each expression.
Rationalize the denominator
Answer

29. Rationalizing the Denominator of Radicals Expressions

30. Rationalizing the Denominator of Radicals Expressions
If the denominator contains a radical and it is not a monomial term, then the use of a conjugate is required.
conjugate

31. Simplify
Use the conjugate

32. Rationalizing the Denominator of Radicals Expressions
conjugate

33. Solving Radical Equations
To solve a radical equation involving one radical:
Isolate the radical expression on one side of the equation.
Raise both sides of the equation to the power that is the same as the order of the radical (inverse operation).
Solve the resulting equation for the variable.
Check for extraneous solutions* by checking the apparent solutions in the original equation.
*Extraneous solutions may be introduced when both sides of an equation are raised to an even power.

34. Radical Equations and Problem Solving

35. EXAMPLE 2 2
Square both sides to get rid of the square root ( )

36. RADICAL EQUATIONS ISOLATE RADICAL / RATIONAL RAISE BOTH SIDES
TO RECIPROCAL POWER
SOLVE FOR THE VARIABLE

37. Radical Equations and Problem Solving

38. Radical Equations and Problem Solving

39. To solve a radical equation involving two square roots:
Isolate one of the radical expressions on one side of the equation.
Square both sides of the equation.
Simplify.
Isolate the remaining radical expression on one side of the equation.
Solve the resulting equation for the variable.
Check for extraneous solutions by checking the apparent solutions in the original equation.

40. EXAMPLE ( ) 2 2 NO SOLUTION Since 16 doesn’t plug in as a solution.
Note: You will get Extraneous Solutions from time to time – always do a quick check
Let’s Double Check that this works

41. Radical Equations and Problem Solving

42. Radical Equations and Problem Solving

43. Radical Equations and Problem Solving

45. Check: n=2
Since the root of x=2 does not check, it is called an extraneous solution.

46. Check #2: n = 27
The solution is n=27.

48. Check: y=3

49. Solving Radical Equations
Set up the equation
so that there will be
only one radical on
each side of the equal sign.
Solve
Square both sides
of the equation.
Use Foil.
Simplify.
Simplify by dividing
by a common factor of 2.
Square both sides of the
equation.
Use Foil.

50. x - 3 = 0 or x - 7 = 0 x = 3 or x = 7 Solving Radical Equations
Distribute the 4.
Simplify.
Factor the quadratic.
Solve for x.
x - 3 = 0 or x - 7 = 0
x = 3 or x = 7
Verify both solutions.
L.S R.S.
L.S R.S.