Exponents for Seventh Graders

πŸ†Practice powers (for 7th grade)

Exponents are a shorthand way of telling us that a number is multiplied by itself.
The number that is multiplied by itself is called the base. The base is the larger number on the left.
The smaller number on the right tells us how many times the number is multiplied by itself. It is called the exponent, or power.

We will usually read it as (base) to the power of (exponent), OR (base) to the (exponent) power.

For example, in the expression 434^3

4 is the base, while 3 is the exponent.
The exponent tells us the number of times the base is to be multiplied by itself.
In our example, 4 (the base) is multiplied by itself 3 times (the exponent): 4Γ—4Γ—4 4\times4\times4
We can call this 4 to the power of 3, or 4 to the third power.

Extra: Since the second and third powers are so common, we have special, short names for them - squared and cubed.

424^2 can be called simply 4 squared.

434^3 can be called simply 4 cubed.


Want to learn more? Check out our videos, examples and exercises on this topic!

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Test yourself on powers (for 7th grade)!

einstein

\( 11^2= \)

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What are powers?

Exponents, or powers, are a shorthand way of writing that a number (the base) is multiplied by itself a certain amount of times (exponent). The exponent itself can be any number. For now, we'll focus on positive, whole numbers.

For example:

  • 424^2 Is called: 'four to the second power' OR 'four to the power of 2' (or 'four squared'). The 4 will be multiplied by another 4 (two fours).
    42=4Γ—4=164^2=4 \times 4=16
  • 434^3 Is called: 'four to the third power' OR 'four to the power of 3' (or 'four cubed'). The 4 will be multiplied three times (three fours).
    43=4Γ—4Γ—4=644^3=4 \times 4 \times 4=64

Remember: The number that is multiplied by itself is called the base, and the number of times the number is multiplied is called the exponent.

Therefore, in the expression 424^2

4 is the base, while 2 is the exponent.
In this case the number 4 is multiplied by itself 2 times, therefore this expression will be called '4 to the second power' OR '4 to the power of 2' OR '4 squared.'

And in the expression 434^3

4 is the base, while 3 is the exponent.
In this case the number 4 is multiplied by itself 3 times, therefore this expression will be called '4 to the third power' OR '4 to the power of 3' or '4 cubed.'


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Exponents in the order of operations

We learned in the Order of Operations that we first solve what's inside the parentheses and then continue with the rest of the equation. What comes next? Let's look at our order of operations acronym: PEMDAS.

The E in PEMDAS is for exponents! They come second and we will solve them after the parentheses and before going on to solve multiplication and division.

Take a moment to memorize the acronym PEMDAS for the order of the mathematical operations:

  1. Parentheses
  2. Exponents
  3. Multiplication and Division
  4. Addition and Subtraction

Multiplying exponents with the same base

anΓ—am=an+m a^n\times a^m=a^{n+m}

When we multiply exponents with the same base, it's like adding the exponents using simple addition.
For example:

  • 45=42+3=42Γ—434^5=4^{2+3}=4^2 \times 4^3
  • X4Γ—X5=X4+5=X9X^4 \times X^5=X^{4+5}=X^9

Do you know what the answer is?

Multiplying several numbers with exponents

(aΓ—bΓ—c)n=anΓ—bnΓ—cn \left(a\times b\times c\right)^n=a^n\times b^n\times c^n

For example:

  • (2Γ—3Γ—5)2=22Γ—32Γ—52(2 \times 3 \times 5)^2 = 2^2 \times 3^2 \times 5^2
  • (xΓ—2Γ—x)2=x2Γ—22Γ—x2(x \times 2 \times x)^2 = x^2 \times 2^2 \times x^2

What happens when the exponents are 0 or 1?

To the power of 0

  • a0 a^0
    Any number to the power of 0 0 is equal to 1 1
  • 0n 0^n
    0 raised to any power (other than zero) is equal to 0 0

(Since 0x0=0, 0x0x0=0 and so forth).

  • 00 0^0
    Zero to the power of zero is undefined.

Check your understanding

To the power of 1

  • 1n=1 1^n=1
    One to any power is one.

(Since 1x1=1, 1x1x1=1 and so forth).

  • x1=x x^1=x
    Any number to the power of 1 1 stays the same number.

Summary


Do you think you will be able to solve it?

Exercises using exponents

  • 62=6^2=
  • 53=5^3=
  • 73=7^3=
  • (13)2=({1\over3})^2=
  • (13)3=({1\over3})^3=
  • (23)3=({2\over3})^3=
  • (32)2=({3\over2})^2=

Example exercises using exponents

Exercise 1

Task:

What exponent will make the following equation true?

7β–‘=49 7^{\square}=49

Solution:

We can find the answer using two different approaches.

Method 1 - Replacement:

We can try replacing the unknown exponent with a number, let's say 2 2 , and check if it fits the equation. In our case it seems that we have arrived at the correct answer.

72=49 7Β²=49

Method 2 - Checking the square root:

49=7 \sqrt{49}=7

That is

72=49 7Β²=49

Answer:

2 2


Test your knowledge

Exercise 2

Task:

210Γ—27Γ—26= 2^{10}\times2^7\times2^6=

Solution:

When multiplying exponents with the same base, we simply add the powers together.

10+7+6=23 10+7+6=23

Therefore, the solution is:

223 2^{23}


Exercise 3

Assignment:

510β‹…57β‹…56= 5^{10}\cdot5^7\cdot5^6=

As we have learned, when multiplying exponents, if the bases of the exponents are the same you can simply add the exponents:

10+7+6=23 10+7+6=23

510β‹…57β‹…56=523 5^{10}\cdot5^7\cdot5^6=5^{23}

Answer:

523 5^{23}


Do you know what the answer is?

Exercise 4

Task:

Solve the following using like terms:

6x6βˆ’9x4=0 6x^6-9x^4=0

Solution:

First, we must clear the smallest power and simplify the numbers by using the common denominator if possible.

6x4(x2βˆ’1.5)=0 6x^4\left(x^2-1.5\right)=0

Separate both sections of the equation that equal to \0\.

We will solve them separately:

6x4=0 6x^4=0

Divide by 6x2 6x^2

x=0 x=0

x2βˆ’1.5=0 x^2-1.5=0

x2=1.5 x^2=1.5

x=Β±32 x=\pm\sqrt{\frac{3}{2}}

Answer:

x=0,x=Β±32 x=0,x=\pm\sqrt{\frac{3}{2}}


Exercise 5: Variables as exponents

Task:

Solve the following equation:

(Am)n (A^m)^n

(4X)2 (4^X)^2

Solution:

(4X)2=4XΓ—2 (4^X)^2=4^{X\times2}

Answer:

4xβ‹…2 4^{x\cdot2}


Check your understanding

Exercise 6

Task:

Simplify the following:

(9ax)4+(4a)x \left(9ax\right)^4+\left(4^a\right)^x

(9Γ—aΓ—x)4+(4a)x= (9\times a\times x)^4+(4^a)^x=

Solution:

We multiply each of the terms in parentheses by its power.

94Γ—a4Γ—x4+4aΓ—x= 9^4\times a^4\times x^4+4^{a\times x}=

94a4x4+4{ax} 9^4a^4x^4+4^{\left\{ax\right\}}

Answer:

94a4x4+4{ax} 9^4a^4x^4+4^{\left\{ax\right\}}


Review questions

What is an exponent?

An exponent, or power, is a simple way to say that a number is multiplied by itself. A power has two elements: a base and an exponent. The exponent tells us the number of times the base is going to be multiplied by itself.

Let's see some examples:

Example 1

34= 3^4=

Here the base is the number 3 3 and the 4 4 is the exponent, which means that the number 3 3 must be multiplied 4 4 times. Then we have the following:

34=3Γ—3Γ—3Γ—3=81 3^4=3\times3\times3\times3=81

Result

34=81 3^4=81


Do you think you will be able to solve it?

Example 2

53= 5^3=

The five must be multiplied by itself three times, so

53=5Γ—5Γ—5=125 5^3=5\times5\times5=125

Result

53=125 5^3=125


Why would I use an exponent?

An exponent can help us to simplify the multiplication of the same number. It is a simple way of indicating the number of times that number should be multiplied by itself.


Test your knowledge

A few important properties of exponents

Let's see a few of the properties of exponents:

  1. The power of 0: Any number raised to the power of 0 is 1 1 .

A0=1 A^0=1

2. The power of 1: Any number raised to the power of 1 will be the same number.

A1=A A^1=A

3. Multiplying powers with the same base:

AmΓ—An=Am+n A^m\times A^n=A^{m+n}

4. Dividing powers with the same base:

AmAn=Amβˆ’n \frac{A^m}{A^n}=A^{m-n}

5. Multiplying powers with the same exponent:

(AΓ—B)n=AnΓ—Bn \left(A\times B\right)^n=A^n\times B^n

6. Dividing powers with the same exponent:

(AB)n=AnBn \left(\frac{A}{B}\right)^n=\frac{A^n}{B^n}

7. Power of a power:

(Am)n=AmΓ—n \left(A^m\right)^n=A^{m\times n}

8. Negative power:

Aβˆ’m=1Am A^{-m}=\frac{1}{A^m}


Using the properties

In order to understand when to use the different properties of exponents, we will need to understand the function of the properties themselves. Let's take a look at some examples:


Example 1

Task:

75Γ—73= 7^5\times7^3=

To solve this problem we will use the third property of exponents: multiplying powers with the same base:

75Γ—73=75+3=78 7^5\times7^3=7^{5+3}=7^8

To solve, we have:

78=7Γ—7Γ—7Γ—7Γ—7Γ—7Γ—7Γ—7=5,764,801 7^8=7\times7\times7\times7\times7\times7\times7\times7=5,764,801

Result

75Γ—73=78 7^5\times7^3=7^8


Example 2

Task:

8684= \frac{8^6}{8^4}=

To solve, we will use the fourth property of powers: dividing powers with the same base:

8684=86βˆ’4=82 \frac{8^6}{8^4}=8^{6-4}=8^2

If we want to simplify the power we will get:

82=8Γ—8=64 8^2=8\times8=64

Result

8684=82 \frac{8^6}{8^4}=8^2


Example 3

Task:

Solve (25)3Γ—2βˆ’3= \left(2^5\right)^3\times2^{-3}=

In the first part we will use the seventh property of powers: power of a power. In the second part we will use the eigth property powers: negative powers.

(25)3=25Γ—3=215 \left(2^5\right)^3=2^{5\times3}=2^{15}

2βˆ’3=123 2^{-3}=\frac{1}{2^3}

Which gives us:

(25)3Γ—2βˆ’3=215Γ—123 \left(2^5\right)^3\times2^{-3}=2^{15}\times\frac{1}{2^3}

Now, we will multiply the fractions

215Γ—123=21523 2^{15}\times\frac{1}{2^3}=\frac{2^{15}}{2^3}

Finally, we will use the fourth property of powers: dividing powers of the same base:

21523=215βˆ’3=212 \frac{2^{15}}{2^3}=2^{15-3}=2^{12}

Result

(25)3Γ—2βˆ’3=212 \left(2^5\right)^3\times2^{-3}=2^{12}


Do you know what the answer is?
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