The exponent of a power

The exponent implies the number of times the base of the power must multiply itself.
In order for the base of the power to know how many times it should multiply itself, we must look at the exponent. The exponent denotes the power to which the base must be raised, that is, it determines how many times we will multiply the base of the power by itself.
How can they remember it?
It is called the exponent because (from the Latin exponentis) it makes visible or exposes how many times the base of the power will be multiplied.
In reality, it not only exposes but also determines.
How will we identify the exponent?
The exponent appears as a small number that is placed in the upper right corner of the base of the power.
It is not the main factor as the base is, therefore, its size is smaller and it appears discreetly to the right side and above it.

Detailed explanation

Start practice

Test yourself on powers!

Choose the expression that is equal to the following:

$$ 2^7 $$

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Let's see the following example:
$a^4$

Could you indicate what the exponent is?
4 of course!
We can clearly see that the exponent is smaller and located at the top right corner of the base of the power.

The number of times that a) must be multiplied by itself is 4.

We can say that:
$a^4=a\times a\times a\times a$
In this example: a) must be multiplied by itself 4 times, as indicated by the exponent.

Exercises on the exponent of a power:

Exercise 1

Assignment

Solve the following exercise:

$(4\times9\times11)^a$

Solution

We will use the formula

$(abc)^m=a^m\times b^m\times c^m$

We solve accordingly

$(4\times9\times11)^a=4^a\times9^a\times11^a=4^a9^a11^a$

Answer

$4^a9^a11^a$

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Test your knowledge

Question 1

$$ 11^2= $$

Question 2

$$ 6^2= $$

Question 3

What is the missing exponent?

$-7^{\square}=-49$

Exercise 2

Prompt

$\left(4^x\right)^y=$

Solution

We multiply the two powers together.

$4^{x\times y}=4^{xy}$

Answer

$4^{xy}$

Exercise 3

Assignment

$x^{-a}=\text{?}$

Solution

$x^{-a}=x^{0-a}$

$\frac{x^0}{x^a}=$

$\frac{1^{}}{x^a}$

Answer

$\frac{1^{}}{x^a}$

Do you know what the answer is?

Question 1

Which of the following is equivalent to the expression below?

$$ $$ 10,000^1 $$

Question 2

Find the value of n:

$6^n=6\cdot6\cdot6$ ?

Question 3

Sovle:

$$ 3^2+3^3 $$

Exercise 4

$2^{-5}=\text{?}$

Solution

$2^{-5}=2^{0-5}=$

$\frac{2^0}{2^5}=$

$\frac{1}{2^5}=$

We solve the exercise in the fraction according to the power

$2^5=2\times2\times2\times2\times2=$

We solve the multiplications from left to right

$4\times2\times2\times2=$

$8\times2\times2=$

$16\times2=32$

Answer

$\frac{1}{32}$

Exercise 5

Assignment

$4^{-1}=\text{?}$

Solution

$4^{-1}=\frac{4^0}{4^1}=$

$\frac{1}{4}$

Answer

$\frac{1}{4}$

Check your understanding

Question 1

$$ 5^3= $$

Question 2

$$ 7^3= $$

Question 3

$\sqrt{x}=2$

Review Questions

What does the exponent in a number's power represent?

The exponent of a base is the number that is found in the upper right part of the base and it is the number that represents or indicates how many times the base should be multiplied by itself.

For example:

$2^4=$

In this power, the base is $2$ and the exponent is $4$ , therefore the exponent indicates that the two should be multiplied by itself $4$ times, that is:

$2^4=\text{ }2\times2\times2\times2$

When a power has no exponent, what number is it?

When a power does not explicitly have an exponent, that is, it lacks an exponent, we must assume that it has an exponent $1$

Examples:

$a=a^1$

$3=3^1$

$7=7^1$

Do you think you will be able to solve it?

Question 1

$\sqrt{x}=6$

Question 2

What is the answer to the following?

$$ 3^2-3^3 $$

Question 3

Which of the following clauses is equal to 100?

What is a power with base one?

In this case, the base will be one, and for this type of power the following holds true:

$1^m=1$

This property tells me that the base one raised to any power will result in $1$ , since one is always multiplied several times, or in this case, the number of times indicated by the exponent.

Examples

$1^3=1\times1\times1=1$

$1^5=1$

$1^8=1$

Test your knowledge

Question 1

Which of the following represents the expression below?

$\frac{1}{5}\cdot\frac{1}{5}\cdot\frac{1}{5}\cdot\frac{1}{5}$ ?

Question 2

Fill in the missing number:

$$ 0^☐=0 $$

Question 3

Choose the expression that is equal to the following:

$$ 2^7 $$

Examples with solutions for The exponent of a power

Exercise #1

Choose the expression that is equal to the following:

$2^7$

Video Solution

Step-by-Step Solution

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

Step 1: Identify the given power expression $2^7$ .
Step 2: Convert $2^7$ 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 $2^7$ is $2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2$ .

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

$2\cdot2\cdot2\cdot2\cdot2\cdot2\cdot2$

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

Answer

$2\cdot2\cdot2\cdot2\cdot2\cdot2\cdot2$

Exercise #2

$11^2=$

Video Solution

Step-by-Step Solution

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

Step 1: Set up the multiplication as $11 \times 11$ .
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 $11 \times 11$ .
Step 2: Perform the multiplication:

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

Let's examine a more structured multiplication method:

Multiply $11$ by $1$ (last digit of the second 11), we get 11.
Multiply $11$ by $10$ (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 $121$ . Upon reviewing the provided choices, the correct answer is choice 4: $121$ .

Therefore, the solution to the problem is $121$ .

Answer

121

Exercise #3

$6^2=$

Video Solution

Step-by-Step Solution

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

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

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

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

Answer

Exercise #4

Which of the following is equivalent to the expression below?

$10,000^1$

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, $10,000^1 = 10,000$ .

Given the choices:

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

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

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

$10,000 \cdot 1$

Answer

$10,000\cdot1$

Exercise #5

Find the value of n:

$6^n=6\cdot6\cdot6$ ?

Video Solution

Step-by-Step Solution

We use the formula: $a\times a=a^2$

In the formula, we see that the power shows the number of terms that are multiplied, that is, two times

Since in the exercise we multiply 6 three times, it means that we have 3 terms.

Therefore, the power, which is n in this case, will be 3.

Answer

$n=3$

Start practice

The exponent of a power

Test yourself on powers!

Exercises on the exponent of a power:

Exercise 1

Exercise 2

Exercise 3

Exercise 4

Exercise 5

Review Questions

What does the exponent in a number's power represent?

When a power has no exponent, what number is it?

What is a power with base one?

Examples with solutions for The exponent of a power

Exercise #1

Video Solution

Step-by-Step Solution

Answer

Exercise #2

Video Solution

Step-by-Step Solution

Answer

Exercise #3

Video Solution

Step-by-Step Solution

Answer

Exercise #4

Video Solution

Step-by-Step Solution

Answer

Exercise #5

Video Solution

Step-by-Step Solution

Answer

More Questions

Powers