The quotient rule for exponents can be used to define negative exponents. It might seem strange to think of negative exponents, but we need to know where they come from and how to work with them.
Factors in the numerator with a negative exponents move to the denominator.
Simplify.
If you are given a factor with a negative exponent in the denominator simply move it to the numerator. Use the following reasoning to justify this.
Simplify.
A common mistake is to multiply the base with the exponent when it is negative. For example,
Avoid this mistake . The correct solution is
Another useful property involves a rational expression raised to a negative exponent.
Simplify.
When simplifying expressions, it usually is best to simplify within the parentheses first and then apply the product and/or the quotient rule.
Simplify.
Scientific notation is an application of negative exponents. It is used to express very large or very small numbers.
An example of a power of ten might look like,
Use this to convert number expressed in scientific notation to a decimal.
Remember that we can obtain this same result by moving the decimal over six places to the right and filling in with the digit 0.
A power of ten might be negative,
Negative exponents appear when working with very small numbers.
Obtain the same result by moving the decimal over to the left 4 places.
Express the number in scientific notation.
Choose to place the decimal so that the first digit is between 1 and 10.
Multiplication is commutative, so when multiplying numbers in scientific notation multiply the decimal parts first. Next, multiply the powers of 10 using the product rule.
In this section we will learn how to divide polynomials. Students find this to be one of the more difficult topics in Algebra. Plan on spending some extra time reviewing the techniques and solutions presented here. (Assume all expressions that appear in a denominator are nonzero.)
Dividing by a Monomial
When dividing we will be using the quotient rule,
This says that when dividing two expressions with the same base you subtract exponents.
In fact, we are using the property for adding fractions with a common denominator
By breaking up the fraction we could then simply and then cancel.
Divide.
A common mistake would be to cancel denominator with only one of the terms. We are dividing the entire expression in the numerator so every term must be cancelled with the denominator.
Dividing by a Polynomial
In this section, we will use polynomial long division when dividing by something other than a monomial. The good news is that the steps are basically the same as the regular division algorithm we are already used to. Use polynomial long division to divide the following.
The completed process follows.
Division does not always work out so evenly, sometimes you will have a remainder.
Divide.
Some of the polynomials will be missing terms. In other words, not all the exponents will be there. When first learning, it really is best to use placeholders and include the missing terms using zero as coefficients.
Divide.
Polynomial long division takes some practice to master. Be patient, do lots of problems and soon you will find them to be enjoyable.
