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Aim: Exponential Equations Course: Alg. 2 & Trig. Aim: How do we solve equations with fractions, negative numbers, or variables in the exponents? Do Now: Meteorologists use the formula D 3 = 216T 2 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?

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Aim: Exponential Equations Course: Alg. 2 & Trig. Model Problem Meteorologists use the formula D 3 = 216T 2 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? D 3 = 216T = 216T 2 substitute D = = 216T 2 simplify and solve 8 = T 2 T 2.8 hours square root of both sides

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Aim: Exponential Equations Course: Alg. 2 & Trig. Power of Power Rule Power of Power Property (a m ) n = a mn (x 2 ) 1/2 = (x -1/2 ) -2 = x 1 = x How do we use this power rule to solve equations like 2x -1/3 = 6 ? x 1 = x raise x -1/3 to the reciprocal power but first....

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Aim: Exponential Equations Course: Alg. 2 & Trig. simplify isolate the variable with the exponent Using the Power of Power Rule Power of Power Property (a m ) n = a mn 2x -1/3 = 6 2 x -1/3 = 3 multiply both sides by the exponents reciprocal (x -1/3 ) -3 = 3 -3 x = 1/3 3 = 1/27 Check your answer but first....

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Aim: Exponential Equations Course: Alg. 2 & Trig. Model Problem Solve for x 3/2 + 1 = 9 isolate variable with a coefficient of 1 raise both members of equation by reciprocal power simplify x 3/2 + 1 = 9 x 3/2 = 8 (x 3/2 ) 2/3 = 8 2/3 x = 8 2/3

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Aim: Exponential Equations Course: Alg. 2 & Trig. Rational Exponents containing variables Solve and check:5 x + 1 = 5 4 for b 0 and b 1, b x = b y x = y because the base on both sides of this equation is 5, we can write the following: x + 1 = 4 x = = 5 4 check: 5 4 = 5 4

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Aim: Exponential Equations Course: Alg. 2 & Trig. Rational Exponents containing variables Solve and check:2 x – 1 = 8 2 check: 2 6 = 8 2 change the right side to base 2 2 x – 1 = (2 3 ) = 8 simplify2 x – 1 = 2 6 equate exponents x – 1 = 6 solve x = 7 64 = – 1 = 8 2 b x = b y x = y

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Aim: Exponential Equations Course: Alg. 2 & Trig. Rational Exponents containing variables Solve and check: check: change both sides to base 2 (2 -2 ) x = (2 3 ) 1 – x 1/4 = 2 -2 simplify2 -2x = 2 3 – 3x equate exponents -2x = 3 – 3x solve x = 3 8 = 2 3

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Aim: Exponential Equations Course: Alg. 2 & Trig. Rational Exponents containing variables Solve and check:9 x + 1 = 27 x check: 9 3 = 27 2 change both sides to base 3 (3 2 ) x + 1 = (3 3 ) x 3 2 = 9 simplify3 2x + 2 = 3 3x equate exponents 2x + 2 = 3x solve x = = = = 27 2

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