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Do Now: Compare the following algebraic expressions: 1. 3x 4 x 2x 2. 3. 4. 3(6x) 4(x y) 0 x and and and and x x x 3 4 6 3 (x ) 4 (xy) x 0 2 Aim: How do we Interpret and Work with Exponents? What is a “power”? Ax Powers sometimes have coefficients BASE b EXPONENT 3x 4 x 2x and (3 4 2)x 9x When adding terms, we combined the COEFFICIENTS. x x x 3 4 x 34 2 x 2 9 When multiplying terms, we combined the EXPONENTS. Product of a Powers Property • When multiplying powers WITH THE SAME BASE, you add their exponents. x x x a In general: b a b Examples: 4 y 3y 12y 6 7 2s 3t s 6s t 6 2 3 9 2 3(6x) 6x + 6x +6x (6+6+6)x 18x Here, the two coefficients were multiplied 6 3 and (x ) x x x 6 6 x 6 666 x 18 Here, the two exponents were multiplied. Power of a Power Property • When a power is being raised to a power, you can simplify the expression by multiplying the two exponents. In general: x m n x m n Examples: a a 7 2 14 3 3 q r qr 4(x y) and 4x 4y (xy) 4 (xy)(xy)( xy)(xy) x x x x y y y y (commutative property of multiplication) the Here, we distributed coefficient to both parts in the parentheses. 4 x y 4 Here, we distributed the exponent to both parts in the parentheses. Power of a Product Property • When a product has an exponent, that exponent is applied to all parts of the product. In general: (xy) x y a a a Examples: (3c) 3 7 7 c 7 (2xy z) 2 x y z 2 3 3 3 6 3 0 x and x x 1 0 0 But WHYYYYYYY?! 02 0 2 2 x x x So, 0 x x x x 0 2 2 Must be the multiplicative identity: 1 Zero Power Property • ANYTHING (besides 0) raised to the 0 power is equal to 1. In general: a 1, a 0 0