In symbols:. In each example we are raising a product to a power. Hence, in each case, we raise each factor to that power. When raising a product of three factors to a power, it is easy to show that we should raise each factor to the indicated power. In general, this is true regardless of the number of factors.
When raising a product to a power, raise each of the factors to the indicated power. In the remaining factors, raising a power to a power requires that we multiply the exponents. The associative and commutative property allows us to multiply all six factors in the order that we please. In this case, we repeat the base and add the exponents.
Raising a quotient to a power is similar to raising a product to a power. However, it is much simpler to realize that when you raise a quotient to a power, you raise both numerator and denominator to that power. To raise a quotient to a power, raise both numerator and denominator to that power. In each example we are raising a quotient to a power.
Hence, in each case, we raise both numerator and denominator to that power. Note that in example c , raising a negative base to an even power produces a positive result.
Can you see that whenever you multiply any two powers of the same base, you end up with a number of factors equal to the total of the two powers? In other words, when the bases are the same , you find the new power by just adding the exponents :.
The rule above works only when multiplying powers of the same base. For instance,. Except in one case: If the bases are different but the exponents are the same , then you can combine them. And it works for any common power of two different bases:. What about dividing? Remember that dividing is just multiplying by 1-over-something. So all the laws of division are really just laws of multiplication.
Well, there are several ways to work it out. However you slice it, you come to the same answer: for division with like bases you subtract exponents , just as for multiplication of like bases you add exponents:. As that example illustrates, you can combine like exponents even when the bases are different:. Use the rule you already know for dividing:.
Write each of these as a single positive power. What do you do with an expression like x 5 4? How many factors of x are there? As you might expect, this applies to any power of a power: you multiply the exponents. In general,. For example,. When division is involved, move variables from the bottom to the top or vice versa to make their exponents positive. Kathryn White has over 11 years of experience tutoring a range of subjects at the kindergarten through college level.
Her writing reflects her instructional ability as well as her belief in making all concepts understandable and approachable. Meijer from Fotolia. Flip fractions with negative exponents in order to make the exponent positive:.
How to Simplify Monomials. How to Simplify Exponents. How to Solve Rational Expression Equations. How to Solve Equations for the Indicated Variable. How to Do Exponents Outside of the Parenthesis. Rules for Multiplying Scientific Notation. How to Factor Polynomials With Fractions. How to Square a Fraction With a Variable.
How to Classify Polynomials by Degree.
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