Home >> Science & Nature >> Science

Rules for Multiplying With Negative Exponents

An exponent denotes how many times the base number should be multiplied by itself. For example, 6^4 is equivalent to 6 * 6 * 6 * 6. The base can also be a variable, as with x^3, which equals x * x * x. When multiplying negative exponents, you must first apply the rule of negative exponents and then use the rules that apply to the multiplication of any integer exponent.

Negative Exponent Rule
- When presented with a negative exponent in the form x^-a, create an inverse with the exponential expression at the bottom with the exponent now positive. For example, x^-4 becomes 1 / (x^4). This also works when the base is given: 3^-2 = 1 / (3^2) = 1 / 9. If the original negative exponent is given as part of an inverse, such as 1 / (x^-3), then the answer is simply the base raised to the positive exponent: 1 / (x^-3) = 1.
Product Rule for Exponents
- The product role for exponents states that the multiplication of two exponential expressions with like bases but differing exponents results in the like base raised to the addition of the exponents. In positive exponents, this would follow the form x^a * x^b = x^(a + b). The same form is used with negative exponents, except that the answer needs to be put into inverse form. For example, x^-3 * x^-4 = x^(-3 + -4) = x^-7 = 1 / (x^7). An example with a given base: 3^-2 * 3^-9 = 3^(-2 + -9) = 3^(-11) = 1 / (3^11).
Power Rule for Exponents
- The power rule for exponents states that when an exponential expression is within parenthesis and the parenthesis is raised to another exponent, the result is the base raised to the multiplication of the two exponents. In positive numbers, this follows the form (x^a)^b = x^(a * b). If only the interior exponent is negative, simply follow the form for the positive numbers and then create the inverse. For example, (x^-3)^4 = x^(-3 * 4) = x^-12 = 1 / (x^12). But if both exponents are negative, the multiplication results in a positive so the inverse isn't needed. For example, (2^-2)^-3 = 2^(-2 * -3) = 2^6 = 64.
Products to Powers Rule
- The products to powers rule states that when two terms are multiplied within parenthesis and raised to a single exterior exponent, the result is each interior term raised to that exponent. For positive exponents, this follows the form (xy)^a = x^a * y^a. If the interior multiplication involves a variable and the exponent is negative, create the inverse of each term for the answer and simplify. For example, (3x)^-2 becomes 1 / (3^2) * 1 / (x^2), which simplifies to (1/9) * (1/x^2) or 1 ( 9x^2). If the interior contains two numbers, create the inverses first and then multiply the answer. For example, (2 * 3) ^-3 becomes (1 / 2^3) * (1 / 3^3) = (1 / 8) * (1 /27) = 1 / 216.

Science