What is an exponent?

Powers are the number that is multiplied by itself several times.
Each power consists of two main parts: 

  • Base of the power: The number that fulfills the requirement of duplication. The principal number is written in large size.
  • Exponent: the number that determines how many times the power base needs to be multiplied by itself.
    The exponent is written in small size and appears on the right side above the power base.
A - How we will identify the exponent

Practice Exponents Rules

Examples with solutions for Exponents Rules

Exercise #1

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

3432=342=32 \frac{3^4}{3^2}=3^{4-2}=3^2 Therefore, the correct answer is option b.

Answer

32 3^2

Exercise #2

192=? 19^{-2}=\text{?}

Video Solution

Step-by-Step Solution

In order to solve the exercise, we use the negative exponent rule.

an=1an a^{-n}=\frac{1}{a^n}

We apply the rule to the given exercise:

192=1192 19^{-2}=\frac{1}{19^2}

We can then continue and calculate the exponent.

1192=1361 \frac{1}{19^2}=\frac{1}{361}

Answer

1361 \frac{1}{361}

Exercise #3

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=bmn \frac{b^m}{b^n}=b^{m-n}

We apply it in the problem:

2423=243=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

Exercise #4

9993= \frac{9^9}{9^3}=

Video Solution

Step-by-Step Solution

Note that in the fraction and its denominator, there are terms with the same base, so we will use the law of exponents for division between terms with the same base:

bmbn=bmn \frac{b^m}{b^n}=b^{m-n}

Let's apply it to the problem:

9993=993=96 \frac{9^9}{9^3}=9^{9-3}=9^6

Therefore, the correct answer is b.

Answer

96 9^6

Exercise #5

(4274)2= (\frac{4^2}{7^4})^2=

Video Solution

Step-by-Step Solution

(4274)2=42×274×2=4478 (\frac{4^2}{7^4})^2=\frac{4^{2\times2}}{7^{4\times2}}=\frac{4^4}{7^8}

Answer

4478 \frac{4^4}{7^8}

Exercise #6

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

Video Solution

Step-by-Step Solution

To begin with, we must remind ourselves of the Negative Exponent rule:

an=1an a^{-n}=\frac{1}{a^n} We apply it to the given expression :

1123=123 \frac{1}{12^3}=12^{-3} Therefore, the correct answer is option A.

Answer

123 12^{-3}

Exercise #7

(43)2= (4^3)^2=

Step-by-Step Solution

To solve (43)2 (4^3)^2 , we use the power of a power rule which states that (am)n=amn (a^m)^n = a^{m \cdot n} .

Here, a=4 a = 4 , m=3 m = 3 , and n=2 n = 2 .

So, we calculate 432 4^{3 \cdot 2} ,

which simplifies to 46 4^6 .

Answer

46 4^6

Exercise #8

(35)4= (3^5)^4=

Video Solution

Step-by-Step Solution

To solve the exercise we use the power property:(an)m=anm (a^n)^m=a^{n\cdot m}

We use the property with our exercise and solve:

(35)4=35×4=320 (3^5)^4=3^{5\times4}=3^{20}

Answer

320 3^{20}

Exercise #9

(23)6= (2^3)^6 =

Step-by-Step Solution

To solve the given expression (23)6 (2^3)^6 , we apply the power of a power rule (am)n=amn (a^m)^n = a^{m \cdot n} . Here, a=2 a = 2 , m=3 m = 3 , and n=6 n = 6 .

Thus, we calculate the exponent:

36=18 3 \cdot 6 = 18

So, (23)6=218 (2^3)^6 = 2^{18} .

Answer

218 2^{18}

Exercise #10

(2×7×5)3= (2\times7\times5)^3=

Video Solution

Step-by-Step Solution

To solve the problem, we need to apply the power of a product exponent rule. This rule states that when you raise a product to a power, it's the same as raising each factor to that power. In mathematical terms, if you have (abc)n (abc)^n , it is equivalent to an×bn×cn a^n \times b^n \times c^n .

Let's apply this rule step by step:

Our original expression is: (2×7×5)3 (2 \times 7 \times 5)^3 .

We first identify the factors inside the parentheses as 2 2 , 7 7 , and 5 5 .

According to the Power of a Product rule, we can distribute the exponent3 3 to each factor:

First, raise 2 2 to the power of 3 3 to get 23 2^3 .

Then, raise 7 7 to the power of 3 3 to get 73 7^3 .

Finally, raise 5 5 to the power of 3 3 to get 53 5^3 .

Therefore, the expression (2×7×5)3 (2 \times 7 \times 5)^3 simplifies to 23×73×53 2^3 \times 7^3 \times 5^3 .

Answer

23×73×53 2^3\times7^3\times5^3

Exercise #11

54×25= 5^4\times25=

Video Solution

Step-by-Step Solution

To solve this exercise, first we note that 25 is the result of a power and we reduce it to a common base of 5.

25=5 \sqrt{25}=5 25=52 25=5^2 Now, we go back to the initial exercise and solve by adding the powers according to the formula:

an×am=an+m a^n\times a^m=a^{n+m}

54×25=54×52=54+2=56 5^4\times25=5^4\times5^2=5^{4+2}=5^6

Answer

56 5^6

Exercise #12

Choose the expression that is equal to the following:

27 2^7

Video Solution

Step-by-Step Solution

To solve this problem, we'll focus on expressing the power 27 2^7 as a series of multiplications.

  • Step 1: Identify the given power expression 27 2^7 .
  • Step 2: Convert 27 2^7 into a product of repeated multiplication. This involves writing 2 multiplied by itself for a total of 7 times.
  • Step 3: The expanded form of 27 2^7 is 2×2×2×2×2×2×2 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 .

By comparing this expanded form with the provided choices, we see that the correct expression is:

2222222 2\cdot2\cdot2\cdot2\cdot2\cdot2\cdot2

Therefore, the solution to the problem is the expression that matches this expanded multiplication form, which is the choice 1: 2222222\text{1: } 2\cdot2\cdot2\cdot2\cdot2\cdot2\cdot2.

Answer

2222222 2\cdot2\cdot2\cdot2\cdot2\cdot2\cdot2

Exercise #13

62= 6^2=

Video Solution

Step-by-Step Solution

To solve this problem, we'll follow these steps:

  • Step 1: Recognize that 62 6^2 means 6×6 6 \times 6 .
  • Step 2: Perform the multiplication of 6 by itself.

Now, let's work through each step:
Step 1: The expression 62 6^2 indicates we need to multiply 6 by itself.
Step 2: Calculating 6×6 6 \times 6 gives us 36.

Therefore, the value of 62 6^2 is 36.

Answer

36

Exercise #14

112= 11^2=

Video Solution

Step-by-Step Solution

To solve this problem, we'll follow these steps:

  • Step 1: Set up the multiplication as 11×11 11 \times 11 .
  • Step 2: Compute the product using basic arithmetic.
  • Step 3: Compare the result with the provided multiple-choice answers to identify the correct one.

Now, let's work through each step:
Step 1: We begin with the calculation 11×11 11 \times 11 .
Step 2: Perform the multiplication:

  1. Multiply the units digits: 1×1=1 1 \times 1 = 1 .
  2. Next, for the tens digits: 11×10=110 11 \times 10 = 110 .
  3. Add the results: 110+1=111 110 + 1 = 111 . This doesn't seem right, so let's break it down further.

Let's examine a more structured multiplication method:

Multiply 11 11 by 1 1 (last digit of the second 11), we get 11.
Multiply 11 11 by 10 10 (tens place of the second 11), we get 110.

If we align correctly and add the partial products:

     11
+   110
------------
   121

Step 3: The correct multiplication yields the final result as 121 121 . Upon reviewing the provided choices, the correct answer is choice 4: 121 121 .

Therefore, the solution to the problem is 121 121 .

Answer

121

Exercise #15

Which of the following is equivalent to the expression below?

10,0001 10,000^1

Video Solution

Step-by-Step Solution

To solve this problem, we will apply the rule of exponents:

  • Any number raised to the power of 1 remains unchanged. Therefore, by the identity property of exponents, 10,0001=10,000 10,000^1 = 10,000 .

Given the choices:

  • 10,00010,000 10,000 \cdot 10,000 : This is 10,0002 10,000^2 .
  • 10,0001 10,000 \cdot 1 : Simplifying this expression yields 10,000, which is equivalent to 10,0001 10,000^1 .
  • 10,000+10,000 10,000 + 10,000 : This results in 20,000, not equivalent to 10,0001 10,000^1 .
  • 10,00010,000 10,000 - 10,000 : This results in 0, not equivalent to 10,0001 10,000^1 .

Therefore, the correct choice is 10,0001 10,000 \cdot 1 , which simplifies to 10,000, making it equivalent to 10,0001 10,000^1 .

Thus, the expression 10,0001 10,000^1 is equivalent to:

10,0001 10,000 \cdot 1

Answer

10,0001 10,000\cdot1