Simplifying Real Numbers With Exponents

Learning Outcomes

  • Simplify expressions with exponents and integer bases
  • Simplify expressions with exponents and rational bases

Remember that an exponent indicates 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]. This format is known as exponential notation.

Exponential Notation

On the left side, a raised to the m is shown. The m is labeled in blue as an exponent. The a is labeled in red as the base. On the right, it says a to the m means multiply m factors of a. Below this, it says a to the m equals a times a times a times a, with m factors written below in blue.
This is read [latex]a[/latex] to the [latex]{m}^{\mathrm{th}}[/latex] power.

In the expression [latex]{a}^{m}[/latex], the exponent tells us how many times we use the base [latex]a[/latex] as a factor.

On the left side, 7 to the 3rd power is shown. Below is 7 times 7 times 7, with 3 factors written below. On the right side, parentheses negative 8 to the 5th power is shown. Below is negative 8 times negative 8 times negative 8 times negative 8 times negative 8, with 5 factors written below.
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]

Solution

1.
[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]

 

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example

Simplify:

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

 

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example

Simplify:

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

 

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