Given a rational expression, the quotient of two polynomials, we will factor the numerator and denominator if we can and then cancel factors that are exactly the same.
When evaluating rational expressions, plug in the appropriate values either before simplifying or after, the result will be the same. Although, it is more efficient to simplify first then evaluate.
We can see that when evaluating, the result will be the same whether or not we simplify first. It turns out that not all numbers can be used when we evaluate.
The point is that not all real numbers will be defined in the above rational expression. In fact, there are two restrictions to the domain, -2 and 3/5. These values, when plugged in, will result in zero in the denominator. Another way to say this is that the domain consists of all real numbers except for −2 and 3/5.
Tip: To find the restrictions, set each factor in the denominator equal to zero and solve. The factors in the numerator do not contribute to the list of restrictions.
Simplify and state the restrictions to the domain.
Even if the factor cancels it still contributes to the list of restrictions. Basically, it is important to remember the domain of the original expression when simplifying. Also, we must use caution when simplifying, please do not try to take obviously incorrect shortcuts like this:
Since subtraction is not commutative, we must be alert to opposite binomial factors. For example, 5 − 3 = 2 and 3 − 5 = −2. In general,
Simplify and state the restrictions to the domain.
At this point, we evaluate using function notation.
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.
