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

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Test yourself on powers!

What is the missing exponent?

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

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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

Choose the expression that is equal to the following:

$$ 2^7 $$

Question 2

Which of the following is equivalent to the expression below?

$$ $$ 10,000^1 $$

Question 3

$$ 11^2= $$

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

$$ 6^2= $$

Question 2

Which of the following clauses is equal to 100?

Question 3

Which of the following represents the expression below?

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

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

Find the value of n:

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

Question 2

What is the answer to the following?

$$ 3^2-3^3 $$

Question 3

Sovle:

$$ 3^2+3^3 $$

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

$$ 7^3= $$

Question 2

$$ 5^3= $$

Question 3

$\sqrt{x}=6$

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

$\sqrt{x}=2$

Question 2

$(\frac{1}{2})^2=$

Question 3

What is the missing exponent?

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

Examples with solutions for The exponent of a power

Exercise #1

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

$n=3$

Exercise #2

What is the answer to the following?

$3^2-3^3$

Video Solution

Step-by-Step Solution

Remember that according to the order of operations, exponents come before multiplication and division, which come before addition and subtraction (and parentheses always before everything),

So first calculate the values of the terms in the power and then subtract between the results:

$3^2-3^3 =9-27=-18$ Therefore, the correct answer is option A.

Answer

$-18$

Exercise #3

Sovle:

$3^2+3^3$

Video Solution

Step-by-Step Solution

Remember that according to the order of operations, exponents precede multiplication and division, which precede addition and subtraction (and parentheses always precede everything).

So first calculate the values of the terms in the power and then subtract between the results:

$3^2+3^3 =9+27=36$ Therefore, the correct answer is option B.

Answer

Exercise #4

Solve the following exercise and circle the correct answer:

$5^2-4^2+2^2=$

Video Solution

Step-by-Step Solution

To solve the expression $5^2 - 4^2 + 2^2$ , we'll need to follow the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right)). Here, we only have exponents and basic arithmetic.

First, calculate the powers:
$5^2 = 25$ ,
$4^2 = 16$ ,
$2^2 = 4$ .

Now substitute the calculated values back into the expression:
$25 - 16 + 4$ .

Perform the subtraction and addition from left to right:
$25 - 16 = 9$ .

Then add $4$ to $9$ :
$9 + 4 = 13$ .

The final answer is $13$ .

Answer

Exercise #5

Solve the following exercise and circle the correct answer:

$4^2-4^3=$

Video Solution

Step-by-Step Solution

To solve the expression $4^2 - 4^3$ , we start by evaluating each power separately:

Calculate $4^2$ :
$4^2$ means $4$ multiplied by itself, which is $4 \times 4 = 16$ .
Calculate $4^3$ :
$4^3$ means $4$ multiplied by itself three times, which is $4 \times 4 \times 4 = 64$ .

Next, substitute these values back into the expression:

$4^2 - 4^3 = 16 - 64$

Perform the subtraction:

$16 - 64 = -48$

Thus, the correct answer is $-48$ .

Answer

-48

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