Definition of Exponentiation

Exponents are a way to write the multiplication of a term by itself several times in a shortened form.

The number that is multiplied by itself is called the base, while the number of times the base is multiplied is called the exponent.

A - Exponentiation

an=aβ‹…aβ‹…a a^n=a\cdot a\cdot a ... (n times)

For example:

5β‹…5β‹…5β‹…5=54 5\cdot5\cdot5\cdot5=5^4

5 5 is the base, while 4 4 is the exponent.

In this case, the number 5 5 is multiplied by itself 4 4 times and, therefore, it is expressed as 5 5 to the fourth power or 5 5 to the power of 4 4 .

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Test yourself on exponents rules!

einstein

\( 112^0=\text{?} \)

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The rules of exponentiation are rules that help us perform operations like addition, subtraction, multiplication, and division with powers. In certain exercises, if the rules of exponentiation are not used correctly, it will be very difficult to find the solution, so it's important to know them.

Don't worry! These aren't complicated rules. If you make an effort to understand them and practice enough, you'll be able to apply them easily.


Understanding Exponents

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Multiplication of Powers with the Same Base

anβ‹…am=an+m a^n\cdot a^m=a^{n+m}

If we multiply powers with the same base, the exponent of the result will be equal to the sum of the exponents.

For example

52β‹…53=52+3=55 5^2\cdot5^3=5^{2+3}=5^5

7X+1β‹…72X+2=73X+3 7^{X+1}\cdot7^{2X+2}=7^{3X+3}

X4β‹…X5=X4+5=X9X^4\cdot X^5=X^{4+5}=X^9


Do you know what the answer is?

Division of Powers with the Same Base

anam=anβˆ’m \frac{a^n}{a^m}=a^{n-m}

a≠0 a≠0

When we divide powers with the same base, the exponent of the result will be equal to the difference of the exponents.

For example

5453=54βˆ’3=51 \frac{5^4}{5^3}=5^{4-3}=5^1

72X7X=72Xβˆ’X=7X \frac{7^{2X}}{7^X}=7^{2X-X}=7^X

X7X5=X7βˆ’5=X2 \frac{X^7}{X^5}=X^{7-5}=X^2


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Powers of Powers

Let's look at the following example of power of a power

(an)m=anβ‹…m \left(a^n\right)^m=a^{n\cdot m}

When we come across a power of a power, the result will be the multiplication of those powers.


For example

(a2)3=a2β‹…3=a6 \left(a^2\right)^3=a^{2\cdot3}=a^6

(aX)2=a2X \left(a^X\right)^2=a^{2X}


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Power of the Multiplication of Several Terms

(aβ‹…bβ‹…c)n=anβ‹…bnβ‹…cn \left(a\cdot b\cdot c\right)^n=a^n\cdot b^n\cdot c^n

For example

(2β‹…3β‹…4)2=22β‹…32β‹…42 \left(2\cdot3\cdot4\right)^2=2^2\cdot3^2\cdot4^2

(Xβ‹…2β‹…X)2=X2β‹…22β‹…X2 \left(X\cdot2\cdot X\right)^2=X^2\cdot2^2\cdot X^2

(X2β‹…2β‹…y3)2=X4β‹…22β‹…y6 \left(X^2\cdot2\cdot y^3\right)^2=X^4\cdot2^2\cdot y^6


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Fractional Exponents

(ab)n=anbn (\frac{a}{b})^n=\frac{a^n}{b^n}


For example

(53)2=5232 (\frac{5}{3})^2=\frac{5^2}{3^2}

(Xy)3=X3y3 (\frac{X}{y})^3=\frac{X^3}{y^3}


Do you know what the answer is?

Negative Exponents

Let's look at the following example of a negative exponent

aβˆ’n=1an a^{-n}=\frac{1}{a^n}

1aβˆ’n=an \frac{1}{a^{-n}}=a^n

This rule is often used to get rid of negative exponents.


For example

5βˆ’2=152=125 5^{-2}=\frac{1}{5^2}=\frac{1}{25}

12βˆ’3=23=8 \frac{1}{2^{-3}}=2^3=8


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Rules for Raising 0 to a Power

a0=1 a^0=1

Any number raised to the power of 0 0 equals 1 1 .

0n=0 0^n=0

The number 0 0 raised to any power (other than 0 0 ) equals 0 0 .

00 0^0 = undefined

The value of the number 0 0 raised to the power of 0 0 is undefined.

Rules About Raising 1 to Any Power

1n=1 1^n=1

The number 1 1 raised to any power is equal to 1 1 .

Do you think you will be able to solve it?

Power Exercises

Exercise 1: (Variables in the value of the power)

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}


Exercise 2: (Number of Elements)

(am)n=amΓ—n (a^m)n^=a^{m\times n}

Task:

Solve the exercise:

(X2Γ—3)2=? (XΒ²\times3)Β²=\text{?}

Solution:

(X2Γ—3)2=X2Γ—2Γ—32=X4Γ—9=9X4(XΒ²\times3)Β²=X^{2\times2}\times3Β²=X^4\times9=9X^4

Before the formula:

(aΓ—b)m=amΓ—bm (a\times b)^{m=}a^m\times b^m

And also

(am)n= (a^m)^n= Power of a power

Answer:

9X4 9X^4


Test your knowledge

Exercise 3

Task:

Solve the exercise:

((7β‹…3)2)6+(3βˆ’1)3β‹…(23)4=? ((7\cdot3)^2)^6+(3^{-1})^3\cdot(2^3)^4=\text{?}

Solution:

(7β‹…3)2β‹…6+3βˆ’1β‹…3β‹…23β‹…4=? (7\cdot3)^{2\cdot6}+3^{-1\cdot3}\cdot2^{3\cdot4}=\text{?}

2112+3βˆ’3β‹…212=? 21^{12}+3^{-3}\cdot2^{12}=\text{?}

Answer:

2112+3βˆ’3β‹…212 21^{12}+3^{-3}\cdot2^{12}

For problems like the following, you can use the formula:

(am)n=amβ‹…n (a^m)^n=a^{m\cdot n}


Exercise 4: (Properties of Powers)

Task:

Solve the following equation:

23β‹…24+(43)2+2523= 2^3\cdot2^4+(4^3)^2+\frac{2^5}{2^3}=

Solution:

23β‹…24+(43)2+2523= 2^3\cdot2^4+(4^3)^2+\frac{2^5}{2^3}=

23+4+43β‹…2+2(5βˆ’3)=27+46+22 2^{3+4}+4^{3\cdot2}+2^{(5-3)}=2^7+4^6+2^2

Answer:

27+46+22 2^7+4^6+2^2

The answer is supported by a series of properties:

  1. (am)n=amβ‹…n (a^m)^n=a^{m\cdot n}
  2. aXaY=aXβˆ’Y \frac{a^X}{a^Y}=a^{X-Y}
  3. aXβ‹…aY=aX+Y a^X\cdot a^Y=a^{X+Y}

Do you know what the answer is?

Exercise 5

Task:

Which expression has the greatest value?

102,24,37,55 10^{2},2^{4},3^{7},5^{5}

Solution:

102=100 10^2=100

24=16 2^4=16

37=2187 3^7=2187

55=3125 5^5=3125

Answer:

The greatest value is 55 5^5


Exercise 6

Task:

Solve the following equation:

((4X)3Y)2 ((4X)^{3Y})^2

Solution:

(4X)3Yβ‹…2=4X6Y (4X)^{3Y\cdot2}=4X^{6Y}


Check your understanding

Exercise 7

Formula:

(am)n=amβ‹…n (a^m)^n=a^{m\cdot n}

Assignment:

Solve the following equation:

(42)3+(93)4=? (4^2)^3+(9^3)^4=\text{?}

Solution:

(42)3+(93)4=? (4^2)^3+(9^3)^4=\text{?}

42β‹…3+93β‹…4=46+912 4^{2\cdot3}+9^{3\cdot4}=4^6+9^{12}

Answer:

46+912 4^6+9^{12}


Questions and Answers on the Topic of Exponentiation

What are the laws of exponents?

Multiplication with the same bases, division with the same bases, and power of powers.


How is multiplication with the same bases done?

The exponents are added.


How is division with the same bases done?

The exponents are subtracted.


What is a number raised to the 0 equal to?

One, as long as the base is not zero.


Do you think you will be able to solve it?

examples with solutions for exponents rules

Exercise #1

79Γ—7= 7^9\times7=

Video Solution

Step-by-Step Solution

According to the property of powers, when there are two powers with the same base multiplied together, the exponents should be added.

According to the formula:anΓ—am=an+m a^n\times a^m=a^{n+m}

It is important to remember that a number without a power is equivalent to a number raised to 1, not to 0.

Therefore, if we add the exponents:

79+1=710 7^{9+1}=7^{10}

Answer

710 7^{10}

Exercise #2

82β‹…83β‹…85= 8^2\cdot8^3\cdot8^5=

Video Solution

Step-by-Step Solution

All bases are equal and therefore the exponents can be added together.

82β‹…83β‹…85=810 8^2\cdot8^3\cdot8^5=8^{10}

Answer

810 8^{10}

Exercise #3

1123=? \frac{1}{12^3}=\text{?}

Video Solution

Step-by-Step Solution

First, we recall the power property for a negative exponent:

aβˆ’n=1an a^{-n}=\frac{1}{a^n} We apply it to the expression we obtained:

1123=12βˆ’3 \frac{1}{12^3}=12^{-3} Therefore, the correct answer is option A.

Answer

12βˆ’3 12^{-3}

Exercise #4

8132= \frac{81}{3^2}=

Video Solution

Step-by-Step Solution

First, we recognize that 81 is a power of the number 3, which means that:

34=81 3^4=81 We replace in the problem:

8132=3432 \frac{81}{3^2}=\frac{3^4}{3^2} Keep in mind that the numerator and denominator of the fraction have terms with the same base, therefore we use the property of powers to divide between terms with the same base:

bmbn=bmβˆ’n \frac{b^m}{b^n}=b^{m-n} We apply it in the problem:

3432=34βˆ’2=32 \frac{3^4}{3^2}=3^{4-2}=3^2 Therefore, the correct answer is option b.

Answer

32 3^2

Exercise #5

2423= \frac{2^4}{2^3}=

Video Solution

Step-by-Step Solution

Let's keep in mind that the numerator and denominator of the fraction have terms with the same base, therefore we use the property of powers to divide between terms with the same base:

bmbn=bmβˆ’n \frac{b^m}{b^n}=b^{m-n} We apply it in the problem:

2423=24βˆ’3=21 \frac{2^4}{2^3}=2^{4-3}=2^1 Remember that any number raised to the 1st power is equal to the number itself, meaning that:

b1=b b^1=b Therefore, in the problem we obtain:

21=2 2^1=2 Therefore, the correct answer is option a.

Answer

2 2

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