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!

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

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

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 #3

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 #4

In the figure in front of you there are 3 squares

Write down the area of the shape in potential notation

Video Solution

Step-by-Step Solution

Using the formula for the area of a square whose side is b:

$S=b^2$ In the picture, we are presented with three squares whose sides from left to right have a length of 6, 3, and 4 respectively:

Therefore the areas are:

$S_1=3^2,\hspace{4pt}S_2=6^2,\hspace{4pt}S_3=4^2$ square units respectively,

Consequently the total area of the shape, composed of the three squares, is as follows:

$S_{\text{total}}=S_1+S_2+S_3=3^2+6^2+4^2$ square units

To conclude, we recognise through the rules of substitution and addition that the correct answer is answer C.

Answer

$6^2+4^2+3^2$

Exercise #5

Choose the expression that is equal to the following:

$2^7$

Video Solution

Answer

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

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