Chapter 3: Variables, Expressions and Equations

3.2.2: Exponential Expressions

Learning Outcomes

  • Simplify expressions using the Product Property of Exponents
  • Simplify expressions using the Power Property of Exponents
  • Simplify expressions using the Product to a Power Property of Exponents

Key words

  • Product Property of Exponents: to multiply exponential terms with the same base, add the exponents.
  • Power Property of Exponents: to raise a power to a power, multiply the exponents.
  • Product to a Power Property of Exponents: to raise a product to a power, raise each factor to that power.

The Product Property of Exponents

In 2.4.1, we derived the properties of exponents for multiplication using integers. This property also works for variables.

For example, [latex]\color{blue}{x^4}\cdot\color{green}{x^5}=\color{blue}{x\cdot x\cdot x\cdot x}\cdot\color{green}{x\cdot x\cdot x\cdot x\cdot x}=x^9[/latex].

We restate the product property here.

The Product Property OF Exponents

For any real number [latex]x[/latex] and any rational numbers [latex]m[/latex] and [latex]n[/latex], [latex]\left(x^{m}\right)\left(x^{n}\right) = x^{m+n}[/latex].

To multiply exponential terms with the same base, add the exponents.

example

Simplify: [latex]{x}^{5}\cdot {x}^{7}[/latex]

Solution

[latex]{x}^{5}\cdot {x}^{7}[/latex]
Use the product property, [latex]{a}^{m}\cdot {a}^{n}={a}^{m+n}[/latex]. [latex]x^{\color{red}{5+7}}[/latex]
Simplify. [latex]{x}^{12}[/latex]

Example

Simplify.

[latex](a^{3})(a^{7})[/latex]

Solution

The base of both exponents is [latex]a[/latex], so the product rule applies.

[latex]\left(a^{3}\right)\left(a^{7}\right)[/latex]

Add the exponents with a common base.

[latex]a^{3+7}[/latex]

Answer

[latex]\left(a^{3}\right)\left(a^{7}\right) = a^{10}[/latex]

try it

 

example

Simplify: [latex]{b}^{4}\cdot b[/latex]

Solution

[latex]{b}^{4}\cdot b[/latex]
Rewrite, [latex]b={b}^{1}[/latex]. [latex]{b}^{4}\cdot {b}^{1}[/latex]
Use the product property, [latex]{a}^{m}\cdot {a}^{n}={a}^{m+n}[/latex]. [latex]b^{\color{red}{4+1}}[/latex]
Simplify. [latex]{b}^{5}[/latex]

try it

try it

 

We can extend the Product Property of Exponents to more than two factors.

example

Simplify: [latex]{x}^{3}\cdot {x}^{4}\cdot {x}^{2}[/latex]

Solution

[latex]{x}^{3}\cdot {x}^{4}\cdot {x}^{2}[/latex]
Add the exponents, since the bases are the same. [latex]x^{\color{red}{3+4+2}}[/latex]
Simplify. [latex]{x}^{9}[/latex]

try it

 

The following video shows more examples of how to use the product rule for exponents to simplify expressions.

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Caution! Do not try to apply this rule to sums.

 

 

Think about the expression [latex]\left(2+3\right)^{2}[/latex]. Does [latex]\left(2+3\right)^{2}[/latex] equal [latex]2^{2}+3^{2}[/latex]?

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

[latex]\left(2+3\right)^{2}=5^{2}=25[/latex]

and

[latex]2^{2}+3^{2}=4+9=13[/latex]

Therefore, you can only use this rule when the numbers inside the parentheses are being multiplied (or divided, as we will see next).

The Power Property of Exponents

In 2.4.1, we derived the properties of exponents for multiplication. This property also works for variables.

For example, [latex]\left ( \color{blue}{x^4}\right )^{5}=\color{blue}{x^4}\color{blue}{x^4}\color{blue}{x^4}\color{blue}{x^4}\color{blue}{x^4}=x^{20}[/latex].

We restate the power property here.

Power Property of Exponents

If [latex]x[/latex] is a real number and [latex]a,b[/latex] are whole numbers, then

[latex]{\left({x}^{a}\right)}^{b}={x}^{a\cdot b}[/latex]

To raise a power to a power, multiply the exponents.

example

Simplify:

1. [latex]{\left({x}^{5}\right)}^{7}[/latex]

2. [latex]{\left({3}^{6}\right)}^{8}[/latex]

Solution

1.
[latex]{\left({x}^{5}\right)}^{7}[/latex]
Use the Power Property, [latex]{\left({a}^{m}\right)}^{n}={a}^{m\cdot n}[/latex]. [latex]x^{\color{red}{5\cdot{7}}}[/latex]
Simplify. [latex]{x}^{35}[/latex]
2.
[latex]{\left({3}^{6}\right)}^{8}[/latex]
Use the Power Property, [latex]{\left({a}^{m}\right)}^{n}={a}^{m\cdot n}[/latex]. [latex]3^{\color{red}{6\cdot{8}}}[/latex]
Simplify. [latex]{3}^{48}[/latex]

try it

Example

Simplify [latex]6\left(c^{4}\right)^{2}[/latex].

Solution

Since we are raising a power to a power, apply the Power Rule and multiply exponents to simplify. The coefficient remains unchanged because it is outside of the parentheses.

[latex]6\left(c^{4}\right)^{2}[/latex]

Answer

[latex]6\left(c^{4\cdot 2}\right)=6c^{8}[/latex]

 

Watch the following video to see more examples of how to use the power rule for exponents to simplify expressions.

The Product to a Power Property

In 2.4.1, we derived the properties of exponents for multiplication.This property also works for variables.

For example, [latex]\left ( \color{blue}{x}\color{green}{y}\right )^{4}=\color{blue}{x\cdot x\cdot x\cdot x}\cdot\color{green}{x\cdot x\cdot x\cdot x}=x^4\,y^4[/latex].

We restate the product to a power property here.

Product to a Power Property of Exponents

If [latex]a[/latex] and [latex]b[/latex] are real numbers and [latex]m[/latex] is a whole number, then

[latex]{\left(ab\right)}^{m}={a}^{m}{b}^{m}[/latex]

To raise a product to a power, raise each factor to that power.

example

Simplify: [latex]{\left(-11x\right)}^{2}[/latex]

Solution

[latex]{\left(-11x\right)}^{2}[/latex]
Use the Power of a Product Property, [latex]{\left(ab\right)}^{m}={a}^{m}{b}^{m}[/latex]. [latex](-11)^{\color{red}{2}}x^{\color{red}{2}}[/latex]
Simplify. [latex]121{x}^{2}[/latex]

try it

 

example

Simplify: [latex]{\left(3xy\right)}^{3}[/latex]

Solution

[latex]{\left(3xy\right)}^{3}[/latex]
Raise each factor to the third power. [latex]3^{\color{red}{3}}x^{\color{red}{3}}y^{\color{red}{3}}[/latex]
Simplify. [latex]27{x}^{3}{y}^{3}[/latex]

try it

 

If the variable has an exponent with it, use the Power Rule: multiply the exponents.

Example

Simplify. [latex]\left(−7a^{4}b\right)^{2}[/latex]

Solution

Apply the exponent 2 to each factor within the parentheses.[latex]\left(−7\right)^{2}\left(a^{4}\right)^{2}\left(b\right)^{2}[/latex]

Square the coefficient and use the Power Rule to square [latex]\left(a^{4}\right)^{2}[/latex].

[latex]49a^{4\cdot2}b^{2}[/latex]

Simplify.

[latex]49a^{8}b^{2}[/latex]

Answer

[latex]\left(-7a^{4}b\right)^{2}=49a^{8}b^{2}[/latex]

 

The next video shows more examples of how to simplify a product raised to a power.

Try It

Simplify.

1. [latex]\left(3x^{5}y\right)^{2}[/latex]

2. [latex]\left(-2a^{3}b\right)^{3}[/latex]

3. [latex]\left(-4y^{5}z^{3}\right)^{2}[/latex]

 

All of the multiplication properties of exponents can be used together and, along with the distributive property, used to simplify algebraic expressions.

Watch the following video for some examples of how to use the power and product rules of exponents to simplify and multiply expressions.

Example

Simplify:  [latex]3x^3\left (4x^2-7x^5\right )[/latex]

Solution

[latex]\color{blue}{3x^3}\left (4x^2-7x^5\right )[/latex]

[latex]=\color{blue}{3x^3}\left (4x^2\right )-\color{blue}{3x^3}\left (7x^5\right )[/latex]         Distribute [latex]3x[/latex] to both terms.

[latex]=3(4)\left (x^3\cdot x^2\right )-3(7)\left ( x^3\cdot x^5\right )[/latex]          Rearrange using the associative and commutative properties.

[latex]=12x^{3+2}-21x^{3+5}[/latex]          Multiply the constants. Add the exponents and keep the common base.

[latex]=12x^{5}-21x^{8}[/latex]

Example

Simplify:  [latex]-2x^3\left (5\left (x^2\right )^4+\left ( 3x^5\right )^2\right )[/latex]

Solution

[latex]-2x^3\left (5\left (x^2\right )^4+\left ( 3x^5\right )^2\right )[/latex]

[latex]=-2x^3\left (5x^8+\left ( 3x^5\right )^2\right )[/latex]

[latex]=-2x^3\left (5x^8+9x^{10}\right )[/latex]

[latex]=-2x^3\left (5x^8 \right )-2x^3\left (9x^{10}\right )[/latex]

[latex]=-2(5)\left (x^3\cdot x^8 \right )-2(9)\left (x^3\cdot x^{10}\right )[/latex]

[latex]=-10x^{11}-18x^{13}[/latex]

 

Try It

Simplify:  [latex]-5x^2\left (3x^4-\left ( 2x^5\right )^3\right )[/latex]

try it

try it

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