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3. Integers & Exponents
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United States of AmericaVirginia
Grade 6

3.03 Introduction to exponents

Lesson
Practice

Introduction to exponents

An exponent (or power) is a small number placed in the upper right hand corner of another number to note how many times a base is being multiplied by itself.

For example, in the expression 10^{3} the number 10 is the base term and the number 3 is the exponent (or index or power). The expression 10^{3} is the same as 10\cdot10\cdot10, or the number 10 multiplied 3 times.

10 raised to the power of 3 is equal to 10x10x10. 10 is labeled 'base', 3 is labeled 'exponent or power',and 10x10x10 is labeled 'multiplied by itself 3 times'.

In the above expression, we call 10^{3} the exponential form and 10\cdot10\cdot10 the expanded form of the expression.

We often encounter a power of 2 when measuring area. Consider the area of a square, for example, which is given by side length times side length. A number, e.g. 5 with an exponent (or power) of 2, can be expressed as 5^{2}, and can be read as "5 to the power of 2" or "five squared".

A number, e.g. 10 to the power of 3, can be expressed as 10^{3}, and can be read as "ten cubed". A power of 3 is involved in calculations like measuring the volume of a cube.

x squared illustrated as a square having a side length of x which means x times x is equals to x squared. x cubed illustrated as a cube having a side length of x which means x times x times x is equals to x cubed.

A base to the power of any other number, e.g. 3^4, can be read as "three to the power of four", and means that the base number is multiplied by itself the number of times shown in the exponent.

\displaystyle 3^4\displaystyle =\displaystyle 3\cdot3\cdot3\cdot3

To evaluate or simplify an exponential expression, the only step we need to take is completing the multiplication.

\displaystyle 3^{4}\displaystyle =\displaystyle 3\cdot3\cdot3\cdot3
\displaystyle =\displaystyle 81Simplify the multiplication

Exploration

Complete the following table of values using a pattern:

2^{0}2^{1}2^{2}2^{3}2^{4}
48

  1. Describe the pattern you used to complete the table.

  2. What do you notice about 2^{1}?

  3. What do you notice about 2^{0}?

  4. Test this observation by filling in a new table with a different base. Do you notice the same thing?

  5. Now try to complete the entire table if the base is 1. What do you notice?

Any number raised to the power of 1 is equal to the original number. And 1 raised to any power is still 1 because 1 times itself any number of times will always be 1.

Any number raised to the power of 0 is 1. Though there is debate among mathematicians about whether 0^0=1 or is undefined.

Examples

Example 1

Identify the base of 3^{2}.

Worked Solution
Create a strategy

Use the base and exponent definition: \text{base}^{\text{exponent}}

Apply the idea
\displaystyle \text{base}^{\text{exponent}}\displaystyle =\displaystyle 3^{2}

The base of the expression is 3.

Example 2

Identify the exponent of 4^{6}.

Worked Solution
Create a strategy

Use the base and exponent definition: \text{base}^{\text{exponent}}

Apply the idea
\displaystyle \text{base}^\text{exponent}\displaystyle =\displaystyle 4^{6}

The exponent of the expression is 6.

Example 3

Write 7^{5} \cdot 6^{4} in expanded form.

Worked Solution
Create a strategy

Use the exponent to know how many times the base should be multiplied by itself.

Apply the idea
\displaystyle 7^{5} \cdot 6^{4}\displaystyle =\displaystyle 7 \cdot 7 \cdot 7 \cdot 7 \cdot 7 \cdot 6 \cdot 6 \cdot 6 \cdot 6Multiply each of the bases by themselves the number of times indicated by the exponent

Example 4

Write 8 \cdot 8 \cdot 8 \cdot 8 \cdot 8 in exponential form:

Worked Solution
Create a strategy

To write repeated multiplication of the same number in exponential form, count how many times the number is multiplied by itself. This will be the exponent. The base will be the number that is multiplied repeatedly.

Apply the idea

The number 8 is multiplied by itself 5 times, so in exponential form, this is written as 8^{5}.

Example 5

Given the table of values:

Exponential formExpanded formEvaluate
4^1
4^2
4^3
4^4
4^5
4^6
a

Complete the table of values

Worked Solution
Create a strategy

The expanded form shows the base being multiplied by itself repeatedly, the number of times equivalent to the exponent. To evaluate, we calculate the result of this multiplication.

Apply the idea
Exponential formExpanded formEvaluate
4^144
4^24\cdot 416
4^34\cdot4\cdot464
4^44\cdot4\cdot4\cdot4256
4^54\cdot4\cdot4\cdot4\cdot41024
4^64\cdot4\cdot4\cdot4\cdot4\cdot44096
b

What do you notice about the numbers in the "Evaluate" column?

Worked Solution
Create a strategy

Observe the pattern formed by the numbers in the "Evaluate" column to identify any relationships or sequences.

Apply the idea

Each number in the "Evaluate" column is four times the number before it. This pattern reflects the fact that as the exponent grows larger by 1 we are multiplying by 4 an additional time.

Reflect and check

We will continue to explore this concept of exponential growth throughout our mathematics courses. It demonstrates how quickly values can grow as the exponent increases, which has many real world applications.

Idea summary

An exponent (or power) notes how many times a base is being multiplied by itself.

A base to the power of any other number means that the base number is multiplied by itself the number of times shown in the exponent.

Outcomes

6.NS.3

The student will recognize and represent patterns with whole number exponents and perfect squares.

6.NS.3a

Recognize and represent patterns with bases and exponents that are whole numbers.

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