Numerical Expressions

Lesson 6.01

After completing this lesson, you will be able to say:

• I can write numerical expressions involving whole-number exponents.

• I can evaluate numerical expressions involving whole-number exponents.

• I can solve order of operation expressions that contain exponents.

Key Terms

• Exponential form: A number including a base and an exponent.

• Base: The number that is multiplied by itself when written in exponential form.

• Exponent:A number that is written above and to the right of a base to indicate how many times to multiply the

base by itself; sometimes called a power

Exponential form

Exponential form is just a simplified way of writing a multiplication expression where a number is being multiplied by itself

Writing a Number in Exponential form

Area in Exponential Form

Since the 5 is being multiplied by itself 2 times, you can use an exponent of 2. The area 5 ft × 5 ft written in exponential form is 52 ft2.

When the exponent is a 2, this is called squaring the base. So you can say "five squared."

Volume in Exponential Form

To calculate the volume of the circus cube you would multiply 5 ft × 5 ft × 5 ft.

5 is the base, but this time it is multiplied 3 times so the exponent in this case is 3. Therefore, the exponential form of the volume is 53 ft3.

When an exponent is a 3, this is called cubing the base. So you can say "five cubed."

Example using Exponential Form

The goal of this new circus act is for the performers to knock over as many pins as possible. Each pin will knock over three other pins, and each of those will knock over three more pins, and so on.

There are five total rows of pins. The expression to see how many pins to knock down in the fifth row is created by multiplying 3 five times. You can write this expression as 3 × 3 × 3 × 3 × 3 or in exponential form as 35

Try it

Ginger, the circus mouse, gave birth to twins. Each of the twins then gave birth to twins. Then those twins gave birth to twins.

Check your work

To understand how the mice population grew, you would multiply 2 three times.

So 2 × 2 × 2 = 23 or "two cubed."

Reading Exponents

An exponent is sometimes referred to as a power. So 52 can be read as "five to the power of two."

Here are a few other variations for reading exponential expressions:

52 53 54

5 to the second power 5 to the third power 5 to the fourth power

5 to the power of 2 5 to the power of 3 5 to the power of 4

5 squared 5 cubed

5 raised to the second power 5 raised to the third power 5 raised to the fourth power

5 with an exponent of 2 5 with an exponent of 3 5 with an exponent of 4

Typing Exponents

Typing Exponents

An easy way to represent an exponent is to use the ^ symbol (above the number 6 on your keyboard).

So, 53 can be typed as 5^3.

Example: 64 = 6^4

Simplifying exponential numbers

35 = 3 x 3 x 3 x 3 x 3 = 3 x 3 x 3 x 3 x 3

9 x 3 x 3 x 3

27 x 3 x 3

81 x 3

243

When simplifying an exponent, you must remember that 73 = 7 × 7 × 7. It does not equal 7 × 3 or 7·3 or 73

Caution

Try it

Simplify the exponential expression of 6.24. Be sure to round your answer to the nearest tenths place

Check your work

6.24 = 6.2 × 6.2 × 6.2 × 6.2 = 1,477.6336

This is 1,477.6 when rounded to the tenths place

Evaluating Numerical Expressions

When simplifying an expression, you must always follow the order of operations.

Order of operations:The rules of which calculation comes first when evaluating an expression

Simplifying and Expression

Steps to Simplify an expression

Step 1: simplify inside parenthesis

Step 2: simplify the exponents

Step 3: evaluate any multiplication and/or division from left to right

Step 4: complete any addition and/or subtraction from left to right

Example

Try it

Simplify the expression43 ÷ (7 − 3) × 2

Check your work

43 ÷ (7 − 3) × 243 ÷ 4 × 264 ÷ 4 × 2

16 x 232

Now that you completed this lesson, you should be able to say:

• I can write numerical expressions involving whole-number exponents.

• I can evaluate numerical expressions involving whole-number exponents.

• I can solve order of operation expressions that contain exponents.