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Study Guides > Intermediate Algebra

Read: Product and Quotient Rules

Learning Objectives

  • Use the product rule to multiply exponential expressions
  • Use the quotient rule to divide exponential expressions

Use the product rule to multiply exponential expressions

Exponential notation was developed to write repeated multiplication more efficiently. There are times when it is easier or faster to leave the expressions in exponential notation when multiplying or dividing. Let’s look at rules that will allow you to do this. For example, the notation 545^{4} can be expanded and written as 55555\cdot5\cdot5\cdot5, or 625625. And don’t forget, the exponent only applies to the number immediately to its left, unless there are parentheses. What happens if you multiply two numbers in exponential form with the same base? Consider the expression 2324{2}^{3}{2}^{4}. Expanding each exponent, this can be rewritten as (222)(2222)\left(2\cdot2\cdot2\right)\left(2\cdot2\cdot2\cdot2\right) or 22222222\cdot2\cdot2\cdot2\cdot2\cdot2\cdot2. In exponential form, you would write the product as 272^{7}. Notice that 77 is the sum of the original two exponents, 33 and 44. What about x2x6{x}^{2}{x}^{6}? This can be written as (xx)(xxxxxx)=xxxxxxxx\left(x\cdot{x}\right)\left(x\cdot{x}\cdot{x}\cdot{x}\cdot{x}\cdot{x}\right)=x\cdot{x}\cdot{x}\cdot{x}\cdot{x}\cdot{x}\cdot{x}\cdot{x} or x8x^{8}. And, once again, 8 is the sum of the original two exponents. This concept can be generalized in the following way:

The Product Rule for Exponents

For any number x and any integers a and b(xa)(xb)=xa+b\left(x^{a}\right)\left(x^{b}\right) = x^{a+b}. To multiply exponential terms with the same base, add the exponents.

Example

Write each of the following products with a single base. Do not simplify further.
  1. t5t3{t}^{5}\cdot {t}^{3}
  2. (3)5(3)\left(-3\right)^{5}\cdot \left(-3\right)
  3. x2x5x3{x}^{2}\cdot {x}^{5}\cdot {x}^{3}

Answer: Solution Use the product rule to simplify each expression.

  1. t5t3=t5+3=t8{t}^{5}\cdot {t}^{3}={t}^{5+3}={t}^{8}
  2. (3)5(3)=(3)5(3)1=(3)5+1=(3)6{\left(-3\right)}^{5}\cdot \left(-3\right)={\left(-3\right)}^{5}\cdot {\left(-3\right)}^{1}={\left(-3\right)}^{5+1}={\left(-3\right)}^{6}
  3. x2x5x3{x}^{2}\cdot {x}^{5}\cdot {x}^{3}
At first, it may appear that we cannot simplify a product of three factors. However, using the associative property of multiplication, begin by simplifying the first two.
x2x5x3=(x2x5)x3=(x2+5)x3=x7x3=x7+3=x10{x}^{2}\cdot {x}^{5}\cdot {x}^{3}=\left({x}^{2}\cdot {x}^{5}\right)\cdot {x}^{3}=\left({x}^{2+5}\right)\cdot {x}^{3}={x}^{7}\cdot {x}^{3}={x}^{7+3}={x}^{10}
Notice we get the same result by adding the three exponents in one step.
x2x5x3=x2+5+3=x10{x}^{2}\cdot {x}^{5}\cdot {x}^{3}={x}^{2+5+3}={x}^{10}

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Caution! Do not try to apply this rule to sums. Think about the expression (2+3)2\left(2+3\right)^{2}

Does (2+3)2\left(2+3\right)^{2} equal 22+322^{2}+3^{2}?

No, it does not because of the order of operations!

(2+3)2=52=25\left(2+3\right)^{2}=5^{2}=25

and

22+32=4+9=132^{2}+3^{2}=4+9=13

Therefore, you can only use this rule when the numbers inside the parentheses are being multiplied (or divided, as we will see next).
In the following video you will see more examples of using the product rule for exponents to simplify expressions. https://youtu.be/P0UVIMy2nuI In our last product rule example we will show that an exponent can be an algebraic expression.  We can use the product rule for exponents no matter what the exponent looks like, as long as the base is the same.

Example

Multiply. xa+2x3a9x^{a+2}\cdot{x^{3a-9}}

Answer: We have two exponentiated terms with the same base, so we can multiply them together. The product rule for exponents says that we can add the exponents. xa+2x3a9=x(a+2)+(3a9)=x4a7x^{a+2}\cdot{x^{3a-9}}=x^{(a+2)+(3a-9)}=x^{4a-7} The expression can't be simplified any further.

Answer

xa+2x3a9=x4a7x^{a+2}\cdot{x^{3a-9}}=x^{4a-7}

 

Use the quotient rule to divide exponential expressions

Let’s look at dividing terms containing exponential expressions. What happens if you divide two numbers in exponential form with the same base? Consider the following expression.

4542 \displaystyle \frac{{{4}^{5}}}{{{4}^{2}}}

You can rewrite the expression as: 4444444 \displaystyle \frac{4\cdot 4\cdot 4\cdot 4\cdot 4}{4\cdot 4}. Then you can cancel the common factors of 4 in the numerator and denominator: \displaystyle Finally, this expression can be rewritten as 434^{3} using exponential notation. Notice that the exponent, 33, is the difference between the two exponents in the original expression, 55 and 22. So, 4542=452=43 \displaystyle \frac{{{4}^{5}}}{{{4}^{2}}}=4^{5-2}=4^{3}. Be careful that you subtract the exponent in the denominator from the exponent in the numerator. So, to divide two exponential terms with the same base, subtract the exponents.

The Quotient (Division) Rule for Exponents

For any non-zero number x and any integers a and b: xaxb=xab \displaystyle \frac{{{x}^{a}}}{{{x}^{b}}}={{x}^{a-b}}

Example

Write each of the following products with a single base. Do not simplify further.
  1. (2)14(2)9\dfrac{{\left(-2\right)}^{14}}{{\left(-2\right)}^{9}}
  2. t23t15\dfrac{{t}^{23}}{{t}^{15}}
  3. (z2)5z2\dfrac{{\left(z\sqrt{2}\right)}^{5}}{z\sqrt{2}}

Answer: Use the quotient rule to simplify each expression.

  1. (2)14(2)9=(2)149=(2)5\dfrac{{\left(-2\right)}^{14}}{{\left(-2\right)}^{9}}={\left(-2\right)}^{14 - 9}={\left(-2\right)}^{5}
  2. t23t15=t2315=t8\dfrac{{t}^{23}}{{t}^{15}}={t}^{23 - 15}={t}^{8}
  3. (z2)5z2=(z2)51=(z2)4\dfrac{{\left(z\sqrt{2}\right)}^{5}}{z\sqrt{2}}={\left(z\sqrt{2}\right)}^{5 - 1}={\left(z\sqrt{2}\right)}^{4}

As we showed with the product rule, you may be given a quotient with an exponent that is an algebraic expression to simplify.  As long as the bases agree, you may use the quotient rule for exponents.

Example

Simplify. yx3y9x\dfrac{y^{x-3}}{y^{9-x}}

Answer: We have a quotient whose terms have the same base so we can use the quotient rule for exponents. yx3y9x=y(x3)(9x)=y2x12\dfrac{y^{x-3}}{y^{9-x}}=y^{(x-3)-(9-x)}=y^{2x-12}

In the following video, you will we more examples of using the quotient rule for exponents. https://youtu.be/xy6WW7y_GcU

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