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Published byAnna Hogan Modified over 3 years ago

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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?

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**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

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**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

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**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 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

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**Model Problem Solve for x3/2 + 1 = 9 x3/2 + 1 = 9 -1 -1 x3/2 = 8**

isolate variable with a coefficient of 1 x3/ = 8 raise both members of equation by reciprocal power (x3/2)2/3 = 82/3 x = 82/3 simplify

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**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: = 54 54 = 54

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**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

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**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:

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**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: = 272 93 = 272 729 = 729

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Aim: How do we solve quadratic equations by completing the square?

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