7.1: n th roots and rational exponents

7.1: nth roots and rational exponents Objectives: students will be able to… Evaluate nth roots of real numbers using radical notation and rational exponent notation.

nth root of a • If bn = a, then b is the nth root of a • 53= 125, 5 is the cube root of 125 • 24 = 16, 2 is the 4th root of 16 RULE: n is called the “index”. It tells you what root you are taking. EX:

Real nth roots Let n be an integer > 1, and let abe a real number: • If n is odd, then a has 1 nth real root: • If n is even, and a > 0, then a has 2 real nth roots: • If n is even and a = 0, then a has 1 nth root • If n is even and a < 0, then a has no real nth roots no real roots

Find the indicated nth root(s)

Rational exponents • Can rewrite a radical expression with rational exponents. They are one in the same! Example: OR

Using negative exponents

examples Rewrite using rational exponents: Rewrite using radical notation:

Evaluate:

Using your calculator to evaluate Enter 3 ^ (3 / 4)

Solving equations using nth roots Isolate the quantity raised to the nth power, and take the nth root of both sides. Be careful…. If n is odd, 1 solution In n is even, 2 solutions (±)

examples: solve

example A basketball has a volume of about 455.6 in3. The formula for the volume of a basketball is V = 4.18879r3. Find the radius of the basketball.

example The rate r at which an initial deposit P will grow to a balance A in t years with interest compounded n times a year is given by Find r if P = $1000, A = $2000, t= 11 years, and n = 12

7.1: n th roots and rational exponents