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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? Aim: Exponential Equations Course: Alg. 2 & Trig. 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 8 T2 T 2.8 hours Aim: Exponential Equations Course: Alg. 2 & Trig. Power of Power Rule (am)n = am•n Power of Power Property (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 x 1 3 3 1 x but first . . . . Aim: Exponential Equations Course: Alg. 2 & Trig. Using the Power of Power Rule Power of Power Property (am)n = am•n but first . . . . 2x -1/3 = 6 2 2 isolate the variable with the exponent x -1/3 = 3 (x -1/3)-3 = 3-3 x = 1/33 = 1/27 multiply both sides by the exponent’s reciprocal simplify Check your answer Aim: Exponential Equations Course: Alg. 2 & Trig. Model Problem Solve for x3/2 + 1 = 9 x3/2 + 1 = 9 -1 -1 x3/2 (x3/2)2/3 x =8 = 82/3 isolate variable with a coefficient of 1 raise both members of equation by reciprocal power = 82/3 x (3 8)2 4 simplify Aim: Exponential Equations Course: Alg. 2 & Trig. Rational Exponents containing variables 5x + 1 = 5 4 Solve and check: 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 = 5 4 54 = 5 4 Aim: Exponential Equations Course: Alg. 2 & Trig. Rational Exponents containing variables 2x – 1 = 8 2 Solve and check: bx = b y x = y change the right side to base 2 23 = 8 simplify 2x – 1 = (23)2 2x – 1 = 26 equate exponents x –1=6 solve x check: =7 27 – 1 = 8 2 26 = 8 2 64 = 64 Aim: Exponential Equations Course: Alg. 2 & Trig. Rational Exponents containing variables x Solve and check: 1 1 x 8 4 change both sides to base 2 1/4 = 2-2 8 = 23 (2-2)x = (23)1 – x 2-2x = 23 – 3x simplify equate exponents solve -2x x 3 check: 1 81 3 4 Aim: Exponential Equations = 3 – 3x =3 1 1 1 2 8 2 64 8 64 Course: Alg. 2 & Trig. Rational Exponents containing variables 9x + 1 = 27x Solve and check: change both sides to base 3 32 = 9 33 = 27 (32)x + 1 = (33)x 32x + 2 = 33x simplify equate exponents 2x + 2 = 3x solve x check: =2 92 + 1 = 272 93 = 272 729 = 729 Aim: Exponential Equations Course: Alg. 2 & Trig.