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How do you do negative exponents?

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Quick Answer

An expression or variable with negative exponents represents the reciprocal of that expression or variable with the same positive exponent. Write the expression or variable as its reciprocal with the equivalent positive exponent; in other words, write the expression as 1 divided by the expression with its equivalent positive exponent.

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  1. Identify the values in an expression subject to a negative exponent

    In an expression, a negative exponent may apply to a single item or multiple items. For example, in an expression x^(-2), where x is a variable, the negative exponent is applied to the x. In the expression 3x^(-2), the negative exponent is applied to the x, but not the scalar 3. In an expression 6*(3x)^-2, the negative exponent is applied to the 3x, but not the 6.

  2. Manipulate components of an expression with a negative exponent into its reciprocal with positive exponent

    Convert a negative exponent to a positive exponent by rewriting the components of the expression that the negative exponent applies to, writing it instead as its reciprocal with the same positive exponent value. For example, you can write x^(-2) as 1/(x^2). You can write 3x^(-2) as 3/(x^2); you do not take the reciprocal of the 3 because the negative exponent does not get applied to the 3. In an expression 6*(3x)^-2, transform this into 6/((3x)^2), or 6/(9x^2). The same approach applies if multiple variables are in an expression; for example, 3*x^(-5)*y^(-8) becomes 3/(x^5 * y^8). If you have an expression where a component has a negative exponent in the denominator, then you can move the component to the numerator and make the exponent positive. For example, in the expression 5/(2x^(-3)), you can transform this into 5/(2/x^3)) = (5x^3)/2. The same approach applies if performing a calculation on a number using negative exponents. For example, 5^(-4) is equivalent to 1/(5^4) = 1/(5*5*5*5) = 1/625. Similarly, 2/(4^-3) = 2 * (4^3) = 128.

  3. Simplify the expression if possible by combining like terms

    If manipulating negative exponent expressions with variables mentioned in multiple places, you may be able to simplify the result. For example, in an expression 24x^5*(2x)^-3, this becomes 24x^5/((2x)^3) = 24x^5 / (8x^3), or 3*(x^5 / x^3). Remember that when multiplying the same variable, you add the exponents together, and when dividing exponents using the same variable, you subtract the exponents. Therefore, you have 3*x^(5-3) = 3x^2. Take the same approach if calculating negative exponents that only involve numbers. For example, in calculating 10^(-3) / 5^(-2), this becomes 5^2 / 10^3, or 25/1000. Since the greatest common factor of the numerator and denominator is 25, you can simplify it to (25/25) / (1000/25) = 1/40.

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