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How to Deal With Negative Exponents in Rational Expressions

When you're first exposed to exponents, you learn that the exponent tells how many times to multiply the base by itself. So, 6^3 means 6 x 6 x 6. Later this definition is extended to include exponents of zero, fractions and negative numbers. These are not as straightforward and easy to understand, but they follow rational rules that are easy to learn. Negative exponents and rational expressions often go together, and two simple rules will convert all the negative exponents in rational expressions to positive exponents.

Instructions

    • 1

      Use one of the basic laws of exponents to understand how negative exponents work. The rule is x^m / x^n = x^(m - n). This means that if the exponent in the denominator is larger than the exponent in the numerator, the exponent will be negative. For example, x^3 / x^4 = x^(3 - 4) = x^(-1). So x^(-1) = 1/x. Similarly, x^(-2) = 1/x^2 and x^(-3) = 1/x^3. To check this out with some numbers, 2^3 / 2^4 = 8/16 = 1/2. Applying the rule, 2^3 / 2^4 = 2^(-1) = 1/2. You can use the rule x^m / x^n = x^(m - n) to understand how to deal with negative exponents anywhere in a rational expression.

    • 2

      Simplify the effect of a negative exponent in the numerator with the following rule: "Move an expression with a negative exponent in the numerator down to the denominator by making the negative exponent positive." One easy way to see this rule is illustrated when you look at 1 / x^k if you want to get rid of the fraction. Remembering that anything to the zero power is one, you have 1 / X^k = x^0 / x^k = x^(0 - k) = x^(-k).

    • 3

      See what happens to negative exponents in the denominator with the following algebraic derivation: 1/x^(-k) = 1/(1/x^(k)) = (x^k/x^k)(1/(1/x^(k))) = x^k /(x^k/x^k) = x^k / 1 = x^k. This means that the rule for getting rid of negative exponents in the denominator can be expressed as: "Move an expression with a negative exponent in the denominator up to the numerator by making the negative exponent positive." Even more concisely: "When you move an expression with an exponent across the fraction bar, the sign of the exponent changes."

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