Video: Simplifying Monomials - Negative Exponents

Video Transcript

In this video, we will learn how to simplify monomials with negative exponents. We will begin by recalling what we know about positive exponents. We know that two to the fourth power is equal to two multiplied by two multiplied by two multiplied by two. This is equal to 16. Two cubed or two to the third power is equal to two multiplied by two multiplied by two, which is eight. In the same way, we know that two squared is equal to four and two to the first power is two. When we have any expression 𝑥 to the 𝑛th power where 𝑛 is a positive integer, then 𝑛 is telling us the number of times we multiply 𝑥 by itself.

We can hopefully spot a pattern where the exponent is decreasing by one. 16 divided by two is equal to eight, eight divided by two is equal to four, and four divided by two is equal to two. As the exponent decreases by one, the number is divided by two. Following this pattern, we get two to the power of zero is equal to one. We know this is true from our rule of zero exponents, 𝑥 to the power of zero is equal to one. Let’s see what happens when we repeat the pattern for negative exponents, two to the power of negative one, two to the power of negative two, and two to the power of negative three.

We know that one divided by two is equal to one-half. One-half divided by two is one-quarter. And one-quarter divided by two is one-eighth. We might recognize here that eight is equal to two multiplied by two multiplied by two. And four is equal to two multiplied by two. This means that two to the power of negative three appears to be equal to one divided by two cubed. Likewise, two to the power of negative two is equal to one over two squared. This leads us to the general rule for negative exponents that, for any nonzero 𝑥, 𝑥 to the power of negative 𝑛 is equal to one over 𝑥 to the power of 𝑛. We will now look at a simple example involving simplifying exponents to prove this

Simplify two cubed divided by two to the sixth power.

Two cubed or two to the third power is equal to two multiplied by two multiplied by two. Two to the sixth power involves multiplying six twos together. We can divide the numerator and denominator by two three times. This leaves us with one over two multiplied by two multiplied by two or one over two cubed. We know from our quotient rule of exponents that 𝑥 to the power of 𝑎 divided by 𝑥 to the power of 𝑏 is equal to 𝑥 to the power of 𝑎 minus 𝑏. To simplify two cubed divided by two to the sixth power, we can subtract the exponents. Three minus six is equal to negative three. This means that two cubed divided by two to the power of six is equal to two to the power of negative three or one over two cubed. This confirms the rule for negative exponents that 𝑥 to the power of negative 𝑛 is equal to one over 𝑥 to the power of 𝑛 or 𝑥 to the 𝑛th power.

In order to answer the remainder of the questions in this video, we need to recall the five key exponent rules.

The product rule of exponents states that 𝑥 to the power of 𝑎 multiplied by 𝑥 to the power of 𝑏 is equal to 𝑥 to the power of 𝑎 plus 𝑏; we add our exponents. The quotient rule of exponents states that 𝑥 to the power of 𝑎 divided by 𝑥 to the power of 𝑏 is equal to 𝑥 to the power of 𝑎 minus 𝑏. Thirdly, the power rule of exponents states that 𝑥 to the power of 𝑎 raised to the power of 𝑏 is equal to 𝑥 to the power of 𝑎𝑏; we multiply the two exponents. When dealing with zero exponents, we know that 𝑥 to the power of zero equals one. Finally, as just proved, with negative exponents, 𝑥 to the power of negative 𝑛 is equal to one over 𝑥 to the power of 𝑛. We will now look at some questions involving these rules.

Evaluate 14 to the power of negative two.

We recall that our rule for negative exponents states that 𝑥 to the power of negative 𝑛 is equal to one over 𝑥 to the power of 𝑛. This means that 14 to the power of negative two is equal to one over 14 squared. 14 squared is equal to 14 multiplied by 14. There are many ways of calculating this product. One way would be to multiply 14 by 10 and then 14 by four. 14 multiplied by 10 is 140 and 14 multiplied by four is 56. Adding these gives us 196. Therefore, 14 multiplied by 14 or 14 squared is 196. 14 to the power of negative two is equal to one over 196.

Our next question involves the quotient rule of exponents as well as negative exponents.

Evaluate two to the power of negative two divided by two to the power of four.

We recall that our quotient rule of exponents states that 𝑥 to the power of 𝑎 divided by 𝑥 to the power of 𝑏 is equal to 𝑥 to the power of 𝑎 minus 𝑏. This means that two to the power of negative two divided by two to the power of four or two to the fourth power is equal to two to the power of negative two minus four. Negative two minus four is equal to negative six. Our negative rule of exponents states that 𝑥 to the power of negative 𝑛 is equal to one over 𝑥 to the power of 𝑛.

This means that two to the power of negative six is equal to one over two to the sixth power. Two to the sixth power is the same as six twos multiplied together. Two multiplied by two is equal to four. Multiplying this by two gives us eight. Eight multiplied by two is 16. 16 multiplied by two is 32. And 32 multiplied by two is 64. One over two to the sixth power is the same as one over 64. The value of two to the power of negative two divided by two to the power of four is one over 64.

Our next question involves recognizing equivalent expressions.

Which of the following is equivalent to three to the power of negative three? Is it (A) negative nine, (B) negative one-ninth, (C) one-ninth, (D) one twenty-seventh, or (E) 27?

We know that the rule of negative exponents states that 𝑥 to the power of negative 𝑛 is equal to one over 𝑥 to the power of 𝑛. In this question, three to the power of negative three is equal to one over three cubed. We know that three cubed is equal to three multiplied by three multiplied by three. As three multiplied by three is nine and nine multiplied by three is 27, then three cubed equals 27. Three to the power of negative three is therefore equal to one over 27. The correct answer is option (D).

We could have immediately ruled out options (A) and (B) as there is no power that we can raise positive three to which will result in a negative answer. In option (C), we’ve correctly used the reciprocal. However, we’ve multiplied three by three instead of three cubed on the denominator. Option (E) is three cubed and not three to the power of negative three.

Our last two questions are more complicated problems involving more than one of the rules of exponents.

Simplify five 𝑥 to the power of negative eight squared multiplied by six 𝑥 squared squared.

We will begin this question by simplifying each of the parentheses. We recall that 𝑥 to the power of 𝑎 all to the power of 𝑏 is equal to 𝑥 to the power of 𝑎𝑏. Five 𝑥 to the power of negative eight squared can be rewritten as five squared multiplied by 𝑥 to the power of negative eight squared. Five squared is equal to 25. Using the rule of exponents mentioned, we need to multiply negative eight by two. This is equal to negative 16. So the first term simplifies to 25𝑥 to the power of negative 16.

We need to repeat this process for the second part of our expression, six 𝑥 squared all squared. This is the same as six 𝑥 squared multiplied by six 𝑥 squared, which can be written as six squared multiplied by 𝑥 squared squared. Six squared is 36. Multiplying the exponents once again gives us 𝑥 to the fourth power or 𝑥 to the power of four. The original expression is therefore simplified to 25𝑥 to the power of negative 16 multiplied by 36𝑥 to the power of four.

The product rule of exponents states that 𝑥 to the power of 𝑎 multiplied by 𝑥 to the power of 𝑏 is equal to 𝑥 to the power of 𝑎 plus 𝑏. We will use this to simplify 𝑥 to the power of negative 16 multiplied by 𝑥 to the power of four. Negative 16 plus four is equal to negative 12. We also need to multiply 25 by 36. This is equal to 900 as 25 multiplied by 30 is 750 and 25 multiplied by six is 150. Our final answer is 900𝑥 to the power of negative 12.

We will complete this video by looking at one more question.

Simplify 𝑥 to the eighth power to the power of negative two multiplied by 𝑥 to the power of negative six to the power of four divided by 𝑥 to the power of negative eight multiplied by 𝑥 to the power of negative three.

We will begin this question by simplifying the numerator and denominator separately. In order to do this, we recall three of the rules of exponents. The product rule, 𝑥 to the power of 𝑎 multiplied by 𝑥 to the power of 𝑏 is equal to 𝑥 to the power of 𝑎 plus 𝑏. The quotient rule, 𝑥 to the power of 𝑎 divided by 𝑥 to the power of 𝑏 is equal to 𝑥 to the power of 𝑎 minus 𝑏. And the power rule, 𝑥 to the power of 𝑎 to the power of 𝑏 is equal to 𝑥 to the power of 𝑎𝑏.

Let’s begin by using the power rule to simplify 𝑥 to the power of eight to the power of negative two and 𝑥 to the power of negative six to the power of four. In both of these cases, we need to multiply the exponents or powers. Eight multiplied by negative two is negative 16. Negative six multiplied by four is equal to negative 24. So the numerator simplifies to 𝑥 to the power of negative 16 multiplied by 𝑥 to the power of negative 24. We can then use the product rule to simplify the numerator and denominator. Negative 16 plus negative 24 is equal to negative 40. So the numerator simplifies to 𝑥 to the power of negative 40. Negative eight plus negative three is negative 11. So the denominator becomes 𝑥 to the power of negative 11.

Our final step is to use the quotient rule to simplify this expression. We need to subtract the exponents, negative 40 minus negative 11. Subtracting negative 11 is the same as adding 11. Negative 40 plus 11 is equal to negative 29. Our expression simplifies to 𝑥 to the power of negative 29. We could rewrite this expression using our rule for negative exponents. 𝑥 to the power of negative 𝑛 is equal to one over 𝑥 to the power of 𝑛. This means that 𝑥 to the power of negative 29 is also equal to one over 𝑥 to the power of 29 or 𝑥 to the 29th power.

We will now summarize the key points from this video. Negative exponents are defined as follows. For any nonzero 𝑥, we have that 𝑥 to the power of negative 𝑛 is equal to one over 𝑥 to the power of 𝑛. When working with negative exponents as in this video, we may need to use the following rules. We have the product rule of exponents. 𝑥 to the power of 𝑎 multiplied by 𝑥 to the power of 𝑏 is equal to 𝑥 to the power of 𝑎 plus 𝑏. We have the quotient rule. 𝑥 to the power of 𝑎 divided by 𝑥 to the power of 𝑏 is equal to 𝑥 to the power of 𝑎 minus 𝑏. The power rule of exponents states that 𝑥 to the power of 𝑎 raised to the power of 𝑏 is equal to 𝑥 to the power of 𝑎𝑏. Finally, when dealing with zero exponents, 𝑥 to the power of zero is equal to one for any nonzero 𝑥.