Rules of Exponentiation

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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.

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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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.

In this article, we will start by remembering what the definition of power is, and then, we will focus in an orderly manner on the different rules of exponentiation.

What are the laws of exponents?

  • Powers
  • Multiplication of powers with the same base
  • Division of powers with the same base
  • Power of a multiplication
  • Power of a quotient
  • Power of a power
  • Power with zero exponent
  • Powers with a negative integer exponent
  • Taking advantage of all the properties of powers or laws of exponents
  • Exponentiation of integers

In the Tutorela blog, you'll find a variety of articles about mathematics.


Understanding Exponents Exponents are a way to express repeated multiplication of the same number. It's like a shortcut when you want to multiply a number by itself several times. For example, instead of writing 2 x 2 x 2 x 2, we can simply write 2^4. Here, 2 is the base and 4 is the exponent, which tells us how many times we multiply the base by itself. So, 2^4 means 2 multiplied by itself 4 times. Exponents are really useful, especially when dealing with really big numbers!

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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

```html

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}


Do you think you will be able to solve it?

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

```html

(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 When we talk about raising numbers to a power, we often think about multiplying a number by itself a certain number of times. But what happens when the number is 0? This can be a bit tricky, so let's go over the rules. First, let's remember what it means to raise a number to a power. For example, 3^4 means 3 Γ— 3 Γ— 3 Γ— 3, which equals 81. The number 3 is the base, and 4 is the exponent. Now, what about 0? Here are some simple rules to keep in mind: 1. Any non-zero number raised to the power of 0 is 1. For instance, 5^0 = 1. This is because the definition of any number to the power of 0 is set to be 1. 2. Zero raised to any positive exponent is 0. So, 0^3 = 0 Γ— 0 Γ— 0 = 0. This makes sense because you're essentially multiplying 0 by itself several times, and any number multiplied by 0 is 0. 3. However, 0 raised to the power of 0, or 0^0, is considered indeterminate. This means that it doesn't have a clear value. In mathematics, some contexts may define it as 1 for convenience, but this is not universally accepted. Remember these rules when you're working with powers of 0, and you'll be able to handle them like a pro!

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 When we talk about powers or exponents, we're discussing a shortcut for multiplication. For example, 2 to the power of 3 (written as 2^3) means 2 multiplied by itself 3 times: 2 x 2 x 2. But what happens when we raise the number 1 to any power? The answer is simpler than you might think. No matter what exponent you choose, if you raise 1 to that power, the result is always 1. It's like a magic trick that never fails! Here's the rule: 1^n = 1, where "n" can be any numberβ€”positive, negative, or even zero. Yes, that's right, 1 to the power of zero is also 1! This might seem strange, but it's an important rule in mathematics. So, if you see an equation like 1^5, 1^10, or 1^100, remember that the answer will always be 1. It doesn't matter how big or small the exponent is; 1 raised to any power remains 1. It's a handy rule to remember, especially when you're working with exponents in math class.

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}

Homework:

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


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Exercise 3

Homework:

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)

Homework:

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

Homework:

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

Homework:

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

```html

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.


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