Pre - Algebra 25 - Simplifying divided exponential expressions

Hello. I’m Professor Von Schmohawk and welcome to Why U. In the last lecture, we saw how to simplify exponential expressions containing terms which are multiplied together. In this lecture we will see how to simplify exponential expressions with divided terms. We have seen that any number "a" raised to the nth power where n is a positive integer is equivalent to n a's multiplied together. Likewise, "a" to the nth power where n is a negative integer is equivalent to one over n a's multiplied together. So a number with a negative exponent is the same as one over that number with the same positive exponent. We also saw that "a" raised to the zero power is equal to one. That is, unless "a" itself is zero since zero to the zero power is undefined. We also saw that to multiply exponential expressions with the same base we add their exponents. Likewise, to divide exponential expressions with the same base we subtract their exponents. We can use this last rule to simplify exponential expressions with divided terms. For example, let’s simplify x to the fifth power divided by x cubed. Since these two exponential terms have the same base, x we can subtract their exponents which gives us x squared. Now let’s simplify a numerical expression two to the seventeenth power divided by two to the fourteenth power. These two exponential terms have the same base, two so we can subtract their exponents which gives us two cubed. Two cubed, of course is two times two times two which is equal to eight. Notice that subtracting exponents is a lot easier than multiplying seventeen twos in the numerator and fourteen twos in the denominator and then dividing. Now let’s simplify the expression three to the fourth power divided by three to the seventh power. Since the exponent in the denominator is larger than the exponent in the numerator when we subtract the exponents we get a negative exponent three to the negative third power. Remember that a number with a negative exponent is the same as one over that number with the same positive exponent so we can write three to the negative third as one over three cubed which is equal to one over twenty-seven. Next we will simplify the expression y squared divided by y cubed. Subtracting the exponents we get y with an exponent of negative one. This is equal to one over y to the first power which is the same as one over y. Occasionally, you may encounter an exponential expression with a negative exponent in the denominator. Instead of a specific exponent let’s use n to represent any exponent. We can eliminate the negative exponent by moving the term to the numerator and switching the sign of the exponent. To see why this works, instead of one in the numerator let’s write "a" to the zero power, which is the same thing as one. Now we can subtract the exponents and we get an exponent of positive n. So one over "a" to the negative n is equal to "a" to the positive n. This looks a lot like another rule we already know. Any number with a negative exponent is the same as one over that number with the same positive exponent. If we switch the two sides of this equation we can see that both equations are saying the same thing. One over an exponential term is equal to that same term with the sign of its exponent switched. This property is useful for eliminating negative exponents in the denominator of an expression. For example, let’s say that we have an expression which contains a term in the denominator with a negative exponent. We can separate this term from the rest of the expression by rewriting the expression as a product of one over that term times the rest of the expression. We can then rewrite the term with a positive exponent according to our rule and recombine it with the rest of the expression. So we have eliminated the negative exponent by moving the term to the numerator and switching the sign of the exponent to positive. Likewise, we can eliminate a term with a negative exponent in the numerator of an expression. We separate this term from the rest of the expression rewrite it as one over the same term with a positive exponent and recombine it with the rest of the expression. So we have eliminated the negative exponent by moving the term to the denominator. So in an expression written as a fraction we can move any exponential term from the denominator to the numerator or from the numerator to the denominator as long as we switch the sign of its exponent. Now let’s simplify an exponential expression with negative exponents in the numerator and the denominator. One way to simplify this expression is to subtract the exponent of the denominator from the exponent of the numerator. Subtracting negative five is the same as adding five so we get x cubed. If you prefer to work with positive exponents an alternate way of doing this would be to move the numerator term to the denominator and the denominator term to the numerator switching the sign of both exponents to positive. We can then subtract exponents to get x cubed. Now instead of x to the negative two over x to the negative five let’s simplify the expression x to the negative five over x to the negative two. We swap the numerator and denominator terms and switch the sign of the exponents. Then subtracting the exponents we get x to the negative third power which is the same as one over x cubed. So far we have used the rules of exponents to simply exponential expressions with multiplied terms and divided terms as long as the terms have the same base. However, exponential expressions often contain a variety of terms with different bases which are multiplied and divided. In our next lecture, we will learn how to simplify any expression of this type.