# Exponents and Roots

🏆Practice powers and roots

## Exponents and Roots

### What is an exponent?

An exponent tells us the amount of times a number is to be multiplied by itself.

### What is a root?

A root is the inverse operation of exponentiation, which helps us discover which number multiplied by itself gives this result.

The square root is equal to the power of 0.5.

## Test yourself on powers and roots!

What is the missing exponent?

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

## Powers and roots

### What is an exponent?

Exponentiation is the requirement for a number to be multiplied by itself several times.
In other words, when we see a number raised to a certain power, we know that we need to multiply the number by itself several times to reach the actual number.

#### How do you correctly read a number with an exponent?

Let's learn through an example:

Exponent base – The exponent base is the number that is required to be multiplied by itself a certain number of times.
How do we identify it?
The main number written in large – in our example, this number is $4$.

Exponent – The exponent is the number that determines how many times the base is required to be multiplied by itself.
How do we identify it?
The exponent is the small number that appears to the right above the base – in our example, this number is $2$.

We read it like this: $4$ to the power of $2$.

#### How do you solve an exponent?

To solve an exponent, we need to multiply the base of the exponent by itself the number of times the exponent requires us to.

Base of the exponent = $4$
Exponent = $2$

Let's take the base of the exponent and multiply it by itself $2$ times.
We get:
$4*4=16$
and actually -
$4^2=16$

Let's practice another example.

How do you solve the following exponentiation:
$5^3=$

Solution:
Let's understand what the base and the exponent are.
Base of the exponent = $5$
Exponent = $3$
This means we need to multiply $5$ by itself $3$ times.
We get:
$5*5*5=125$
And actually:
$5^3=125$

Another example:

Solve the following power:
$3^3=$

Solution:
At first glance, we see that the base of the exponent and the exponent itself are identical. Does this change anything for us? Not at all, we work according to the rules.
We multiply the number $3$ by itself – for $3$ times and get:
$3*3*3=27$

And actually –
$3^3=27$

Another example:
Solve the following power:
$1^4=$
We need to take the number $1$ and multiply it by itself $4$ times. We get:
$1*1*1*1=1$
What would happen if we had such a power?
$1^{700}=$
Would we really need to write the number $1$ for $700$ times to understand that the result will be $1$ in the end?
No.
From this, we can conclude that: $1$ to the power of any number equals $1$.

Point to Ponder – What happens when the exponent is $1$?
When the exponent is $1$, the number does not change at all and it can be considered as if it has already performed the exponentiation.
For example:
$7^1=7$
Any number to the power of $1$ is the number itself.

Another point to consider – what happens when the exponent is $0$?
When the exponent of the number is $0$, we get a result of $1$. It doesn't matter what the number is.
Any number to the power of $0$ will equal $1$.
That means:
$2^0=1$
$​​​​​​​7^0=1$
${4,675}^0=1$

### What is a root?

A root is equal to the power of $0.5$ and is denoted by the symbol $√$.
We can say that: $\sqrt{a}=a^{0.5}$
A root is the inverse operation of exponentiation.
If a small number appears on the left side, it will be the order of the root.

When any number appears as a regular root, we ask ourselves which number we would need to multiply by itself only twice to get the number inside the root?
In other words, which number raised to the power of $2$ will give us the number that appears inside the root.
For example:
$\sqrt4=2$

If we multiply $2$ by itself twice, we get $4$.

Another example:
$\sqrt16=$

Solution:
If we multiply the number $4$ twice by itself, we get $4$ and therefore:
$\sqrt16=4$

What should you know about roots?

• The result of the square root will always be positive!
You will never get a negative result. We can get a result of $0$.
• There is no answer for $\sqrt{negative~number}$!

It is important to know - roots and exponents take precedence over all four arithmetic operations.
First, we perform the root and exponentiation operations, and only then do we proceed to the order of operations.

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## examples with solutions for powers and roots

### Exercise #1

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 #2

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 #3

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 #4

$\sqrt{441}=$

### Step-by-Step Solution

The root of 441 is 21.

$21\times21=$

$21\times20+21=$

$420+21=441$

$21$

### Exercise #5

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

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

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