Title: P.3 Radicals and Rational Exponents Q: What is a radical? What is a rational number? A: A
1P.3 Radicals and Rational ExponentsQ What is
a radical? What is a rational number?A A
radical involves a root symbol, whereas a
rational number involves a fraction.
2Definition of the Principal Square Root
- If a is a nonnegative real number, the
nonnegative number b such that b2 a, denoted by
b ?a, is the principal square root of a. - That is ?42 (since 2 squared 4), not 2
- (even though (-2) squared also 4).
3Square Roots of Perfect Squares
Ex Simplify ?(-3)2 Ex Simplify ?x2 Ex
Simplify ?-32 Ans 3, x, and not a real
number or 3i
4The Product Rule for Square Roots
- If a and b represent nonnegative real number, then
- The square root of a product is the product of
the square roots.
Ex Compare and draw conclusions ?916 vs.
?916
5- a. ?500
- b. ?6x?3x
- c. ?108x6y11
- Ans a) b) c)
6The Quotient Rule for Square Roots
- If a and b represent nonnegative real numbers and
b does not equal 0, then
- The square root of the quotient is the quotient
of the square roots.
Ex Simplify (ans 10/3)
7Example
We can only add radical expressions if they
contain like terms The same number must be
under the radical sign (the radicand), and it
must have the same index. Then just like
ordinary like terms we add the COEFFICIENTS and
KEEP THE LIKE parts the SAME.
- Ex Perform the indicated operation
- Ans
8- Ex Perform the indicated operation
7?24 2?6 Ans
9Definition of the Principal nth Root of a Real
Number
- If n, the index is
- even, and a is nonnegative (a gt 0) then b is also
nonnegative (b gt 0) - Ex
- odd, a and b can be any real numbers with the
same sign ( or -) - Ex
-
- Q What would we write if n is even and a is
negative? - (Ans not a real number.)
10Finding the nth Roots of Perfect nth Powers
It is only necessary to use the absolute value
symbol if you are finding the even root of a
variable (unknown).
11Ex Simplify each of the following ³?(-2)3 ?(-
2)2 3?-8x7y11 4?16x8y3 5 -x10
32
Ans -2 2
12The Product and Quotient Rules for nth Roots
- For all real numbers, where the indicated roots
represent real numbers,
Q Do you remember for what operation(s) you may
NOT separate ( or reverse to put together) the
numbers? (A sum or difference.)
13Definition of Rational Exponents
The denominator of the rational exponent becomes
the INDEX of the radical expression.
14- Ex Simplify the following
- 4 ½
- (-8)(2/3)
- (250x9y7)1/3
- (125x6)2/3
Ans 2 4
15Definition of Rational Exponents
- The exponent m/n consists of two parts the
denominator n is the root and the numerator m is
the exponent. Furthermore,
16Example Simplify 2(-8x12)-(2/3)
Ans
17Rationalizing the Denominator
ONE TERM in the denominator simplify, then
multiply by whatever is needed to make a perfect
root (ONE TERM). Ex TWO TERMS in the
denominator (one is a square root) simplify,
then multiply by the conjugate (TWO TERMS). Ex
18Simplified form for Radical Expressions
- NO radical sign in the denominator
- NO fractions under the radical sign
- NO exponents greater than the index under the
radical sign - The index is reduced as low as possible