CR.22: Laws of Exponents (Negative Exponents)

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]mnegative rule of exponents to simplify the expression to its reciprocal.

Divide one exponential expression by another with a larger exponent. Use our example, h3h5h3h5.

h3h5=hhhhhhhh=hhhhhhhh=1hh=1h2h3h5=hhhhhhhh=hhhhhhhh=1hh=1h2

If we were to simplify the original expression using the quotient rule, we would have

h3h5=h35=h2h3h5=h35=h2

Putting the answers together, we have h2=1h2h2=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.

an=1an and an=1anan=1an and an=1an

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

an=1an and an=1anan=1an and an=1an

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.

  1. θ3θ10θ3θ10
  2. z2zz4z2zz4
  3. (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.

  1. (3t)2(3t)8(3t)2(3t)8
  2. f47f49ff47f49f
  3. 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.

  1. b2b8b2b8
  2. (x)5(x)5(x)5(x)5
  3. 7z(7z)57z(7z)5

Try It

Write each of the following products with a single base. Do not simplify further. Write answers with positive exponents.

  1. t11t6t11t6
  2. 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.

(pq)3=3 factors(pq)(pq)(pq)=pqpqpq=3 factorsppp3 factorsqqq=p3q3(pq)3=3 factors(pq)(pq)(pq)=pqpqpq=3 factorsppp3 factorsqqq=p3q3

In other words, (pq)3=p3q3(pq)3=p3q3.

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

(ab)n=anbn(ab)n=anbn

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.

  1. (ab2)3(ab2)3
  2. (2t)15(2t)15
  3. (2w3)3(2w3)3
  4. 1(7z)41(7z)4
  5. (e2f2)7(e2f2)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.

  1. (g2h3)5(g2h3)5
  2. (5t)3(5t)3
  3. (3y5)3(3y5)3
  4. 1(a6b7)31(a6b7)3
  5. (r3s2)4(r3s2)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.

(e2f2)7=f14e14(e2f2)7=f14e14

Let’s rewrite the original problem differently and look at the result.

(e2f2)7=(f2e2)7=f14e14 (e2f2)7=(f2e2)7=f14e14 

It appears from the last two steps that we can use the power of a product rule as a power of a quotient rule.

(e2f2)7=(f2e2)7=(f2)7(e2)7=f27e27=f14e14 (e2f2)7=(f2e2)7=(f2)7(e2)7=f27e27=f14e14 

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

(ab)n=anbn(ab)n=anbn

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.

  1. (4z11)3(4z11)3
  2. (pq3)6(pq3)6
  3. (1t2)27(1t2)27
  4. (j3k2)4(j3k2)4
  5. (m2n2)3(m2n2)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.

  1. (b5c)3(b5c)3
  2. (5u8)4(5u8)4
  3. (1w3)35(1w3)35
  4. (p4q3)8(p4q3)8
  5. (c5d3)4(c5d3)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.

  1. (6m2n1)3(6m2n1)3
  2. 175174173
  3. (u1vv1)2
  4. (2a3b1)(5a2b2)
  5. (x22)4(x22)4
  6. (3w2)5(6w2)2

Try It

Simplify each expression and write the answer with positive exponents only.

  1. (2uv2)3
  2. x8x12x
  3. (e2f3f1)2
  4. (9r5s3)(3r6s4)
  5. (49tw2)3(49tw2)3
  6. (2h2k)4(7h1k2)2

In the following video we show more examples of how to find the power of a quotient.