# Basis of a power

The base of the power is the number that is multiplied by itself as many times as indicated by the exponent.
The base appears as a number or algebraic expression. In its upper right corner, the exponent is shown in small.

The base of the power has to stand out clearly since it is the base!

The base of the power can be positive or negative and, depending on the exponent, the sign in the result will be modified.

## Test yourself on powers (for 7th grade)!

$$11^2=$$

Points that are important to remember:

• When the base of the power is a positive number and the exponent is an even number the result will be positive.
• Even when the base of the power is a positive number and the exponent is an odd number, the result will also be positive.
• Even when the base of the power is a negative number and the exponent is an even number, the result will be positive.
• When the base of the power is a negative number and the exponent is odd, the result will be negative.

The base of the power is the number that is multiplied by itself as many times as indicated by the exponent.

How can you remember it?
It is called base because the power is raised on it: it is our base. If the power has no base, then there is no power.
How can we identify the base of the power?
The base of the power will appear as a number or algebraic expression. In its upper right corner we can see, in small, the exponent.
The base of the power has to stand out clearly since it is the base!
Let's see it in the following example:
$a^2$
What is the base of the power?
Of course a!
The base on which we raise the power is a.
In this example the exponent asks a, the base of the power, to multiply by itself twice.
That is:
$a\times a$
We can say that:
$a^2=a\times a$

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## Exercises on the base of a power

### Exercise 1

Prompt

What is the value we will place to solve the following equation?

$7^{\square}=49$

Solution

To answer this question it is possible to answer in two ways:

One way is replacement:

We place power of $2$ and it seems that we have arrived at the correct result, ie:

$7²=49$

Another way is by using the root

$\sqrt{49}=7$

That is

$7²=49$

$2$

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### Exercise 2

Query

What is the result of the following power?

$(\frac{2}{3})^3$

To solve this question we must first understand the meaning of the exercise.

$(\frac{2}{3})\cdot(\frac{2}{3})\cdot(\frac{2}{3})$

$\frac{2}{3}\cdot\frac{2}{3}\cdot\frac{2}{3}$

Now everything is simpler... Correct?

$2\cdot2\cdot2=8$

$3\cdot3\cdot3=27$

We obtain: $\frac{8}{27}$

$\frac{8}{27}$

### Exercise 3

Consigna

$a\cdot b\cdot a\cdot b\cdot a^2$

Solution:

If we break down the exercise we see that it is divided into $2$ coefficients of $a$ and coefficients of $b$

Let's start with the coefficient of $a$

What do we have?

We have $a\cdot a\cdot a²$

That is, we can write this like this:

$a\cdot a\cdot a\cdot a$

This means we can write it like this:

$a^4$

Let's move on to the coefficient $b$.

$b\cdot b=b²$

We add the two together and it turns out that:

$a^4\cdot b^2$

$a^4\cdot b^2$

Do you know what the answer is?

### Exercise 4

Assignment

Solve the exercise:

$\left(a^5\right)^7=$

Solution

We will use the formula

$(a^m)^n=a^{m\times n}$

We multiply the powers together and solve accordingly.

$a^{5\times7}=a^{35}$

$a^{35}$

### Exercise 5

Consigna

$(y\times7\times3)^4=$

Solution

We will use the formula

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

We solve accordingly

$(y\times7\times3)^4=y^4\times7^4\times3^4$

$y^4\times7^4\times3^4$

## Review questions

### How do you read the base and exponent?

In order to read a power, there are special cases such as the power $2$ and $3$.

$a^2$ can be read as: "$a$ to the second power", "$a$ squared" or "$a$ to the power two."

$a^3$ can be read as: "$a$ to the third power", "$a$ cubed".

The others we can read as:

$a^x$ "$a$ raised to the power $x$"

$a^4$,$a$ to the fourth power", " to the fourth power", " to the fifth power", " to the fifth power".

$a^5$,$a$ to the fifth power".

$a^6$: "$a$ to the sixth power" : " to the sixth power".

### What is the base of 3²?

In this example the base is $3$ and the power is the $2$

Do you think you will be able to solve it?

### How is the result of a negative number to an even power and to an odd power?

When we have a negative number and we raise it to a power we can have the following cases:

$\left(-x\right)^{\text{par}}=+$

$\left(-x\right)^{impar}=-$

Let's look at the following examples:

#### Example 1

Calculate the following power

$\left(-2\right)^3$

We can observe that it is a negative number raised to an odd power, therefore the result will be negative, since by the law of signs it is as follows:

$\left(-2\right)^3=\left(-2\right)\left(-2\right)\left(-2\right)=\left(4\right)\left(-2\right)=-8$

$-8$

#### Example 2

Calculate the following power

$\left(-4\right)^4=$

In this example we observe that the power is even, therefore the result will be positive by sign laws, remaining as follows:

$\left(-4\right)^4=\left(-4\right)\left(-4\right)\left(-4\right)\left(-4\right)=\left(16\right)\left(-4\right)\left(-4\right)$

$\left(16\right)\left(-4\right)\left(-4\right)=\left(-64\right)\left(-4\right)=256$

$256$

## examples with solutions for basis of a power

### Exercise #1

What is the answer to the following?

$3^2-3^3$

### 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.

$-18$

### Exercise #2

Sovle:

$3^2+3^3$

### 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.

36

### Exercise #3

Find the value of n:

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

### 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.

$n=3$

### Exercise #4

In the figure in front of you there are 3 squares

Write down the area of the shape in potential notation

### Step-by-Step Solution

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

$S=b^2$In the problem of the drawing, three squares whose sides have a length: 6, 3, and 4, units of length from left to right in the drawing respectively,

Therefore the areas are:

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

Therefore, 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

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

### Exercise #5

$11^2=$