Aim: How do we solve equations with fractions, negative numbers, or variables in the exponents? Do Now: Meteorologists use the formula D3 = 216T2 to describe the size and duration of storms. In the formula, D is the diameter of the storm in miles and T is the duration, or the number of hours the storm lasts. If the diameter of the thunderstorm is 12 miles, about how long would this storm last?

square root of both sides Model Problem Meteorologists use the formula D3 = 216T2 to describe the size and duration of storms. In the formula, D is the diameter of the storm in miles and T is the duration, or the number of hours the storm lasts. If the diameter of the thunderstorm is 12 miles, about how long would this storm last? D3 = 216T2 substitute D = 12 123 = 216T2 simplify and solve 1728 = 216T2 8 = T2 square root of both sides T  2.8 hours

How do we use this power rule to solve Power of Power Rule Power of Power Property (am)n = am•n (x2)1/2 = x1 = x (x -1/2)-2 = x1 = x How do we use this power rule to solve equations like 2x -1/3 = 6 ? raise x -1/3 to the reciprocal power but first . . . .

Using the Power of Power Rule Power of Power Property (am)n = am•n but first . . . . isolate the variable with the exponent 2x -1/3 = 6 2 2 x -1/3 = 3 multiply both sides by the exponent’s reciprocal (x -1/3)-3 = 3-3 simplify x = 1/33 = 1/27 Check your answer

Model Problem Solve for x3/2 + 1 = 9 x3/2 + 1 = 9 -1 -1 x3/2 = 8 isolate variable with a coefficient of 1 -1 -1 x3/2 = 8 raise both members of equation by reciprocal power (x3/2)2/3 = 82/3 x = 82/3 simplify

Rational Exponents containing variables Solve and check: 5x + 1 = 54 for b  0 and b  1, bx = by  x = y because the base on both sides of this equation is 5, we can write the following: x + 1 = 4 x = 3 check: 53 + 1 = 54 54 = 54

Rational Exponents containing variables Solve and check: 2x – 1 = 82 bx = by  x = y change the right side to base 2 2x – 1 = (23)2 23 = 8 simplify 2x – 1 = 26 equate exponents x – 1 = 6 solve x = 7 check: 27 – 1 = 82 26 = 82 64 = 64

Rational Exponents containing variables Solve and check: change both sides to base 2 (2-2)x = (23)1 – x 1/4 = 2-2 8 = 23 simplify 2-2x = 23 – 3x equate exponents -2x = 3 – 3x solve x = 3 check:

Rational Exponents containing variables Solve and check: 9x + 1 = 27x change both sides to base 3 (32)x + 1 = (33)x 32 = 9 33 = 27 simplify 32x + 2 = 33x equate exponents 2x + 2 = 3x solve x = 2 check: 92 + 1 = 272 93 = 272 729 = 729