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1 Math 1 Unit 2, Lesson 4: Properties of Exponents Property: Rule: Example: Zero as an Exponent: a 0 = 1, this says that anything raised to the zero power is 1. Negative Exponent: Multiplying Powers with the Same Base: Raising a Power to a Power: Raising a Product to a Power:, this says that negative exponents in the numerator get moved to the denominator and become positive exponents. Negative exponents in the denominator get moved to the numerator and become positive exponents. Only move the negative exponents. a m a n = a m + n, this says that to multiply two exponents with the same base, you keep the base and add the powers. (a m ) n = a mn, this says that to raise a power to a power you need to multiply the exponents. There are several other rules that go along with the power rule, such as the product-to-powers rule and the quotient-to-powers rule. (ab) m = a m b m, this says that when a product is raised to a power, you need to distribute that power through to each term in the product. (10y) 4 = 10 4 y 4 = 10,000y 4

2 Dividing Powers with the Same Base:, this says that to divide two exponents with the same base, you keep the base and subtract the powers. This is similar to reducing fractions; when you subtract the powers put the answer in the numerator or denominator depending on where the higher power was located. If the higher power is in the denominator, put the difference in the denominator and vice versa, this will help avoid negative exponents. Raising a Quotient to a Power: ( a b )m = am, this says that to raise a quotient to a power bm you apply the power to the numerator and denominator individually ( 5y z )3 = 125y3 z 3

3 Example Problems: Example 1 Simplify: there are no zero powers. Step 4: Apply the Product Rule. ended up in the denominator because there were 10 more x s in the denominator. not reduce.

4 Example 2 Simplify: Step 1: Apply the Zero-Exponent Rule. Step 4: Apply the Product Rule. In this case, the product rule does not apply. Step 5: Apply the Quotient Rule. In this case, the quotient rule does not apply. not reduce.

5 Example 3 Simplify: there are no zero powers. Step 4: Apply the Product Rule. In this case, the product rule does not apply. ended up in the numerator and the y s ended up in the denominator. not reduce.

6 Example 4 Simplify: after applying the zero-exponent rule and multiplying by 1, that term is essentially gone. In this case, I kept the 2 in parentheses because I did not want to lose the negative sign. Step 4: Apply the Product Rule. ended up in the denominator. reduce.

7 Example 5 Simplify: there are no zero powers. Step 4: Apply the Product Rule. In this case, we can apply the rule to the x s and y s in the numerator. ended up in the numerator and the y s ended up canceling out. remaining fractions. In this case, the numbers in the numerator get multiplied together and then the fraction gets reduce.

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