# Evaluating Exponential Expressions

### Learning Outcomes

- Evaluate exponential expressions

## Anatomy of exponential terms

We use exponential notation to write repeated multiplication of the same quantity. For example, [latex]{2}^{4}[/latex] means to multiply four factors of [latex]2[/latex], so [latex]{2}^{4}[/latex] means [latex]2\cdot 2\cdot 2\cdot 2[/latex]. Conversely, [latex]10\cdot10\cdot10[/latex] can be written more succinctly as [latex]10^{3}[/latex]. The [latex]10[/latex] in [latex]10^{3}[/latex]^{ }is called the

**base**. The [latex]3[/latex] in [latex]10^{3}[/latex]

^{ }is called the

**exponent**. The expression [latex]10^{3}[/latex] is called the exponential expression. Knowing the names for the parts of an exponential expression or term will help you learn how to perform mathematical operations on them.

### Exponential Notation

This is read [latex]a[/latex] to the [latex]{m}^{\mathrm{th}}[/latex] power.### Example

Identify the exponent and the base in the following terms, then simplify:- [latex]7^{2}[/latex]
- [latex]{\left(\frac{1}{2}\right)}^{3}[/latex]
- [latex]2x^{3}[/latex]
- [latex]\left(-5\right)^{2}[/latex]

Answer:

1) [latex]7^{2}[/latex] The exponent in this term is [latex]2[/latex] and the base is [latex]7[/latex]. To simplify, expand the term: [latex]7^{2}=7\cdot{7}=49[/latex] 2) [latex]{\left(\frac{1}{2}\right)}^{3}[/latex] The exponent on this term is [latex]3[/latex], and the base is [latex]\frac{1}{2}[/latex]. To simplify, expand the multiplication and remember how to multiply fractions: [latex]{\left(\frac{1}{2}\right)}^{3}=\frac{1}{2}\cdot{\frac{1}{2}}\cdot{\frac{1}{2}}=\frac{1}{8}[/latex] 3) [latex]2x^{3}[/latex] The exponent on this term is [latex]3[/latex], and the base is [latex]x[/latex], the [latex]2[/latex] is not getting the exponent because there are no parentheses that tell us it is. This term is in its most simplified form. 4) [latex]\left(-5\right)^{2}[/latex] The exponent on this terms is [latex]2[/latex] and the base is [latex]-5[/latex]. To simplify, expand the multiplication: [latex]\left(-5\right)^{2}=-5\cdot{-5}=25[/latex]In the following video you are provided more examples of applying exponents to various bases.

https://youtu.be/ocedY91LHKU Before we begin working with variable expressions containing exponents, let’s simplify a few expressions involving only numbers.### example

Simplify: 1. [latex]{5}^{3}[/latex] 2. [latex]{9}^{1}[/latex] Solution1. | |

[latex]{5}^{3}[/latex] | |

Multiply [latex]3[/latex] factors of [latex]5[/latex]. | [latex]5\cdot 5\cdot 5[/latex] |

Simplify. | [latex]125[/latex] |

2. | |

[latex]{9}^{1}[/latex] | |

Multiply [latex]1[/latex] factor of [latex]9[/latex]. | [latex]9[/latex] |

### try it

[ohm_question]146094[/ohm_question]### example

Simplify: 1. [latex]{\left({\Large\frac{7}{8}}\right)}^{2}[/latex] 2. [latex]{\left(0.74\right)}^{2}[/latex]Answer: Solution

1. | |

[latex]{\left({\Large\frac{7}{8}}\right)}^{2}[/latex] | |

Multiply two factors. | [latex]\left({\Large\frac{7}{8}}\right)\left({\Large\frac{7}{8}}\right)[/latex] |

Simplify. | [latex]{\Large\frac{49}{64}}[/latex] |

2. | |

[latex]{\left(0.74\right)}^{2}[/latex] | |

Multiply two factors. | [latex]\left(0.74\right)\left(0.74\right)[/latex] |

Simplify. | [latex]0.5476[/latex] |

### try it

[ohm_question]146095[/ohm_question] [ohm_question]146867[/ohm_question]### example

Simplify: 1. [latex]{\left(-3\right)}^{4}[/latex] 2. [latex]{-3}^{4}[/latex]Answer: Solution

1. | |

[latex]{\left(-3\right)}^{4}[/latex] | |

Multiply four factors of [latex]−3[/latex]. | [latex]\left(-3\right)\left(-3\right)\left(-3\right)\left(-3\right)[/latex] |

Simplify. | [latex]81[/latex] |

2. | |

[latex]{-3}^{4}[/latex] | |

Multiply two factors. | [latex]-\left(3\cdot 3\cdot 3\cdot 3\right)[/latex] |

Simplify. | [latex]-81[/latex] |

^{th}power. In part 2 we raise only the [latex]3[/latex] to the [latex]4[/latex]

^{th}power and then find the opposite.

### try it

[ohm_question]146097[/ohm_question]### Evaluate expressions

Evaluating expressions containing exponents is the same as evaluating the linear expressions from earlier in the course. You substitute the value of the variable into the expression and simplify. You can use the order of operations to evaluate the expressions containing exponents. First, evaluate anything in Parentheses or grouping symbols. Next, look for Exponents, followed by Multiplication and Division (reading from left to right), and lastly, Addition and Subtraction (again, reading from left to right). So, when you evaluate the expression [latex]5x^{3}[/latex] if [latex]x=4[/latex], first substitute the value [latex]4[/latex] for the variable [latex]x[/latex]. Then evaluate, using order of operations.### Example

Evaluate [latex]5x^{3}[/latex] if [latex]x=4[/latex].Answer: Substitute [latex]4[/latex] for the variable [latex]x[/latex].

[latex]5\cdot4^{3}[/latex]

Evaluate [latex]4^{3}[/latex]. Multiply.[latex]5\left(4\cdot4\cdot4\right)=5\cdot64=320[/latex]

#### Answer

[latex-display]5x^{3}=320[/latex] when [latex]x=4[/latex-display]### Example

Evaluate [latex]\left(5x\right)^{3}[/latex] if [latex]x=4[/latex].Answer: Substitute [latex]4[/latex] for the variable [latex]x[/latex].

[latex]\left(5\cdot4\right)3[/latex]

Multiply inside the parentheses, then apply the exponent—following the rules of PEMDAS.[latex]20^{3}[/latex]

Evaluate [latex]20^{3}[/latex].[latex]20\cdot20\cdot20=8,000[/latex]

#### Answer

[latex-display]\left(5x\right)3=8,000[/latex] when [latex]x=4[/latex-display]### Try It

[ohm_question]53024[/ohm_question]### Example

Evaluate [latex]x^{3}[/latex] if [latex]x=−4[/latex].Answer: Substitute [latex]−4[/latex] for the variable x.

[latex]\left(−4\right)^{3}[/latex]

Evaluate. Note how placing parentheses around the [latex]−4[/latex] means the negative sign also gets multiplied.[latex]−4\cdot−4\cdot−4[/latex]

Multiply.[latex]−4\cdot−4\cdot−4=−64[/latex]

#### Answer

[latex-display]x^{3}=−64[/latex] when [latex]x=−4[/latex-display]- [latex]-{3}^{2}[/latex]
- [latex]{\left(-3\right)}^{2}[/latex]

[latex]\begin{array}{c}-\left({3}^{2}\right)\\=-\left(9\right) = -9\end{array}[/latex]

To evaluate 2), you would apply the exponent to the [latex]3[/latex] and the negative sign:

[latex]\begin{array}{c}{\left(-3\right)}^{2}\\=\left(-3\right)\cdot\left(-3\right)\\={ 9}\end{array}[/latex]

The key to remembering this is to follow the order of operations. The first expression does not include parentheses so you would apply the exponent to the integer [latex]3[/latex] first, then apply the negative sign. The second expression includes parentheses, so hopefully you will remember that the negative sign also gets squared.

In the next sections, you will learn how to simplify expressions that contain exponents. Come back to this page if you forget how to apply the order of operations to a term with exponents, or forget which is the base and which is the exponent!

In the following video you are provided with examples of evaluating exponential expressions for a given number.

https://youtu.be/pQNz8IpVVg0## Contribute!

## Licenses & Attributions

### CC licensed content, Original

- Simplify Basic Exponential Expressions.
**Authored by:**James Sousa (Mathispower4u.com) for Lumen Learning.**License:**CC BY: Attribution. - Revision and Adaptation.
**Provided by:**Lumen Learning**License:**CC BY: Attribution. - Screenshot: Repeated Image.
**Provided by:**Lumen Learning**License:**CC BY: Attribution. - Evaluate Basic Exponential Expressions.
**Authored by:**James Sousa (Mathispower4u.com) for Lumen Learning.**License:**CC BY: Attribution.

### CC licensed content, Shared previously

- Unit 11: Exponents and Polynomials, from Developmental Math: An Open Program.
**Provided by:**Monterey Institute of Technology and Education**Located at:**https://www.nroc.org/.**License:**CC BY: Attribution.