Learning Outcomes
- Simplify expressions with negative exponents.
- Simplify exponential expressions.
Using the Negative Rule of Exponents
Another useful result occurs if we relax the condition that m>nm>n in the quotient rule even further. For example, can we simplify h3h5h3h5? When [latex]m
Divide one exponential expression by another with a larger exponent. Use our example, h3h5h3h5.
If we were to simplify the original expression using the quotient rule, we would have
Putting the answers together, we have h−2=1h2h−2=1h2. This is true for any nonzero real number, or any variable representing a nonzero real number.
A factor with a negative exponent becomes the same factor with a positive exponent if it is moved across the fraction bar—from numerator to denominator or vice versa.
We have shown that the exponential expression anan is defined when nn is a natural number, 0, or the negative of a natural number. That means that anan is defined for any integer nn. Also, the product and quotient rules and all of the rules we will look at soon hold for any integer nn.
A General Note: The Negative Rule of Exponents
For any nonzero real number aa and natural number nn, the negative rule of exponents states that
Example: Using the Negative Exponent Rule
Write each of the following quotients with a single base. Do not simplify further. Write answers with positive exponents.
- θ3θ10θ3θ10
- z2⋅zz4z2⋅zz4
- (−5t3)4(−5t3)8(−5t3)4(−5t3)8
Try It
Write each of the following quotients with a single base. Do not simplify further. Write answers with positive exponents.
- (−3t)2(−3t)8(−3t)2(−3t)8
- f47f49⋅ff47f49⋅f
- 2k45k72k45k7
Watch this video to see more examples of simplifying expressions with negative exponents.
Example: Using the Product and Quotient Rules
Write each of the following products with a single base. Do not simplify further. Write answers with positive exponents.
- b2⋅b−8b2⋅b−8
- (−x)5⋅(−x)−5(−x)5⋅(−x)−5
- −7z(−7z)5−7z(−7z)5
Try It
Write each of the following products with a single base. Do not simplify further. Write answers with positive exponents.
- t−11⋅t6t−11⋅t6
- 2512251325122513
Finding the Power of a Product
To simplify the power of a product of two exponential expressions, we can use the power of a product rule of exponents, which breaks up the power of a product of factors into the product of the powers of the factors. For instance, consider (pq)3(pq)3. We begin by using the associative and commutative properties of multiplication to regroup the factors.
In other words, (pq)3=p3⋅q3(pq)3=p3⋅q3.
A General Note: The Power of a Product Rule of Exponents
For any real numbers aa and bb and any integer nn, the power of a product rule of exponents states that
Example: Using the Power of a Product Rule
Simplify each of the following products as much as possible using the power of a product rule. Write answers with positive exponents.
- (ab2)3(ab2)3
- (2t)15(2t)15
- (−2w3)3(−2w3)3
- 1(−7z)41(−7z)4
- (e−2f2)7(e−2f2)7
Try It
Simplify each of the following products as much as possible using the power of a product rule. Write answers with positive exponents.
- (g2h3)5(g2h3)5
- (5t)3(5t)3
- (−3y5)3(−3y5)3
- 1(a6b7)31(a6b7)3
- (r3s−2)4(r3s−2)4
In the following video we show more examples of how to find hte power of a product.
Finding the Power of a Quotient
To simplify the power of a quotient of two expressions, we can use the power of a quotient rule, which states that the power of a quotient of factors is the quotient of the powers of the factors. For example, let’s look at the following example.
Let’s rewrite the original problem differently and look at the result.
It appears from the last two steps that we can use the power of a product rule as a power of a quotient rule.
A General Note: The Power of a Quotient Rule of Exponents
For any real numbers aa and bb and any integer nn, the power of a quotient rule of exponents states that
Example: Using the Power of a Quotient Rule
Simplify each of the following quotients as much as possible using the power of a quotient rule. Write answers with positive exponents.
- (4z11)3(4z11)3
- (pq3)6(pq3)6
- (−1t2)27(−1t2)27
- (j3k−2)4(j3k−2)4
- (m−2n−2)3(m−2n−2)3
Try It
Simplify each of the following quotients as much as possible using the power of a quotient rule. Write answers with positive exponents.
- (b5c)3(b5c)3
- (5u8)4(5u8)4
- (−1w3)35(−1w3)35
- (p−4q3)8(p−4q3)8
- (c−5d−3)4(c−5d−3)4
Simplifying Exponential Expressions
Recall that to simplify an expression means to rewrite it by combing terms or exponents; in other words, to write the expression more simply with fewer terms. The rules for exponents may be combined to simplify expressions.
Example: Simplifying Exponential Expressions
Simplify each expression and write the answer with positive exponents only.
- (6m2n−1)3(6m2n−1)3
- 175⋅17−4⋅17−3
- (u−1vv−1)2
- (−2a3b−1)(5a−2b2)
- (x2√2)4(x2√2)−4
- (3w2)5(6w−2)2
Try It
Simplify each expression and write the answer with positive exponents only.
- (2uv−2)−3
- x8⋅x−12⋅x
- (e2f−3f−1)2
- (9r−5s3)(3r6s−4)
- (49tw−2)−3(49tw−2)3
- (2h2k)4(7h−1k2)2
In the following video we show more examples of how to find the power of a quotient.