Square Root Rules

What is a root?

A square root of a number is a value that, when multiplied by itself, gives the original number. For example, the square root of $25$ is $5$ , because $5 \times 5=25$ .
The square root symbol is written as \sqrt.
for example: $\sqrt{9}=3$

Square root is basically the inverse operation of a power. Even so, it can be written as a power!
Square root is equal to a power of $0.5$ .

If a small number appears to the left of the square root symbol, it indicates the order of the root, also known as the index. For example, in $\sqrt[3]{8}$ , the small number $3$ tells us it is a cube root, meaning we are looking for a number that, when multiplied by itself three times, equals $8$ . When no number is shown, it is understood to be a square root (index of $2$ ).

What is necessary to know about a root?

Perfect Squares:

These are numbers whose square roots are whole numbers, like $1,4,9,16,25$ etc. For example: $\sqrt{16}=4$ because $4×4=16$ .

Non-Perfect Squares:

These are numbers that do not have a whole number square root. For instance, $\sqrt{2}$ is approximately $1.414$ , and it’s an irrational number.

Negative Numbers:

The square root of a negative number is not defined in the set of real numbers. That means the result of the square root will always be positive! You will never get a negative result. We can get a result of $0$ . For $\sqrt{}$ of a negative-number there is no answer! However, in advanced math, we use imaginary numbers (e.g., $\sqrt{-1} = i$ ) to handle these cases.

Square as a Power:

The square root is basically a half power. We can say that: $\sqrt{a}=a^{\frac{1}{2}}$

Roots in order of operations:

A square root precedes the four arithmetic operations. First, perform the square root and only then continue according to the order of operations. When there are both powers and roots, we solve then from the left to the right, since they are on the same level.

Simplifying Square Roots:

When the number inside the square root has a factor that is a perfect square, you can simplify it. For example: $\sqrt{50}=\sqrt{25×2}=\sqrt{25}×\sqrt{2}=5\sqrt2$ .

The result of the square root will always be positive

Detailed explanation

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Test yourself on rules of roots!

Choose the largest value

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What is a square root anyway?

A root is symbolized with the sign $√$
Indeed, when we see a number with a root, we wonder what positive number raised to $2$ , will give us what is written inside the root.
A root is the opposite of a power operation. When there is no small number at the top left of the root, it denotes that it is a root of $2$ , square root.
If a small number appears on the left, this will be the order of the root.

Let's know some of the fundamental laws:

The result of a root will always be positive!! A negative result will never be obtained. We can get a result of $0$ .
For $√$ (negative number) there is no answer!
The root is basically a half power. We can say that: $\sqrt a=a^{ 1 \over 2}$
A root precedes the four arithmetic operations. First, perform the root and only then order the arithmetic operations.

Let's see the example:
$\sqrt {64} =8$

Let's ask, what power of $2$ will give us $64$ and the answer is $8$ .
True, also $-8$ to the power of $2$ will give us $64$ but the result of the root must be positive!

Laws of Radicals

The laws of radicals are very relevant for solving exercises, and combining them with power rules can greatly help you solve exercises easily.

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Test your knowledge

Question 1

Solve the following exercise:

$\sqrt{\frac{2}{4}}=$

Question 2

Solve the following exercise:

$\sqrt{30}\cdot\sqrt{1}=$

Question 3

Solve the following exercise:

$\sqrt{1}\cdot\sqrt{25}=$

The root of a product

When the root appears across the entire product, we can break down each factor and apply the root to them, leaving the multiplication sign between the factors.
We formulate:
$\sqrt{(a\cdot b)}=\sqrt{a}\cdot\sqrt{b}$

Square Root of a Quotient

When the root appears over the entire quotient (over the entire fraction), we can break down each factor and apply the root to it, leaving the division sign (fraction line) between the factors.
We formulate:
$\sqrt{\frac{a}{b}}=\frac{\sqrt{a}}{\sqrt{b}}$

Do you know what the answer is?

Question 1

Solve the following exercise:

$\sqrt{16}\cdot\sqrt{1}=$

Question 2

Solve the following exercise:

$\sqrt{1}\cdot\sqrt{2}=$

Question 3

Solve the following exercise:

$\sqrt{25x^4}=$

Root of a Radical

The root over another root, we will multiply the order of the first root by the order of the second root and the order we obtain will be executed as a root on our number. (As the rule of power over another power)
Let's put it this way:
$\sqrt[n]{\sqrt[m]{a}}=\sqrt[n\cdot m]{a}$

Square Root of a product

When we find a root that is in the entirety of the product, we can break down the product's factors and leave a separate root for each of them.
We will formulate this as a rule:
$\sqrt{(a\cdot b)}=\sqrt{a}\cdot\sqrt{b}$

Let's see this in an example:
$\sqrt{(64\cdot100)}$
According to the root of a product rule, we can break down the factors and leave the root of each factor separately while maintaining the multiplication operation between them:
We will break it down and obtain:
$\sqrt{64}\cdot\sqrt{100}=$
$8\cdot10=80$

Check your understanding

Question 1

Solve the following exercise:

$\sqrt{10}\cdot\sqrt{3}=$

Question 2

Solve the following exercise:

$\sqrt{7}\cdot\sqrt{7}=$

Question 3

Solve the following exercise:

$\sqrt{100}\cdot\sqrt{25}=$

Square Root of a quotient

When we encounter a root that is over the entire quotient (fraction) we can break down the factors of the quotient and leave a separate root for each of them. We will place the division operation between the two factors: the fraction line.

Let's formulate it this way: $\sqrt{\frac{a}{b}}=\frac{\sqrt{a}}{\sqrt{b}}$

Let's see this in an example:

$\sqrt{\frac{36}{9}}$

According to the rule of the root of a quotient, we can break down the factors and leave the root of each factor separately while maintaining the multiplication operation between them:
We will break it down and obtain:

$\frac{\sqrt{36}}{\sqrt{9}}$

$\frac{6}{3}=2$

Square Root of a Radical

When we encounter an exercise where there is a root over a root, we will multiply the order of the first root by the order of the second root and the order we obtain we will multiply as a root over our number. (As in the rule of power over power)
Let's formulate it this way:
$\sqrt[n]{\sqrt[m]{a}}=\sqrt[n\cdot m]{a}$

Let's see this in the example:
$\sqrt[2]{\sqrt[4]{100}}=\sqrt[2\cdot4]{100}=\sqrt[8]{100}$

If you are interested in this article, you may also be interested in the following articles:

The root of a product

Root of the quotient

Radication

Combining powers and roots

In the blog of Tutorela you will find a variety of articles on mathematics.

Examples and exercises with solutions on properties of roots

Exercise #1

Choose the largest value

Video Solution

Step-by-Step Solution

Let's begin by calculating the numerical value of each of the roots in the given options:

$\sqrt{25}=5\\ \sqrt{16}=4\\ \sqrt{9}=3\\$ We can determine that:

$5>4>3>1$ Therefore, the correct answer is option A

Answer

$\sqrt{25}$

Exercise #2

Solve the following exercise:

$\sqrt{\frac{2}{4}}=$

Video Solution

Step-by-Step Solution

Let's simplify the expression, first we'll reduce the fraction under the square root:

$\sqrt{\frac{2}{4}}= \\ \sqrt{\frac{1}{2}}=$

We'll use two exponent laws:

A. Definition of root as a power:

$\sqrt[n]{a}=a^{\frac{1}{n}}$

B. The power law for powers applied to terms in parentheses:

$\big(\frac{a}{b}\big)^n=\frac{a^n}{b^n}$

Let's return to the expression we received, first we'll use the law mentioned in A and convert the square root to a power:

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

We'll continue and apply the power law mentioned in B, meaning- we'll apply the power separately to the numerator and denominator, in the next step we'll remember that raising the number 1 to any power will always give the result 1, and in the fraction's denominator we'll return to the root notation, again, using the power law mentioned in A (in the opposite direction):

$\big(\frac{1}{2}\big)^{\frac{1}{2}}= \\ \frac{1^{\frac{1}{2}}}{2^{\frac{1}{2}}}=\\ \boxed{\frac{1}{\sqrt{2}}}\\$ Let's summarize the simplification of the given expression:

$\sqrt{\frac{2}{4}}= \\ \sqrt{\frac{1}{2}}= \\ \frac{1^{\frac{1}{2}}}{2^{\frac{1}{2}}}=\\ \boxed{\frac{1}{\sqrt{2}}}\\$ Therefore, the correct answer is answer D.

Answer

$\frac{1}{\sqrt{2}}$

Exercise #3

Solve the following exercise:

$\sqrt{30}\cdot\sqrt{1}=$

Video Solution

Step-by-Step Solution

Let's start with a reminder of the definition of a root as a power:

$\sqrt[n]{a}=a^{\frac{1}{n}}$

We will then use the fact that raising the number 1 to any power always yields the result 1, particularly raising it to the power of half of the square root (which we obtain by using the definition of a root as a power mentioned earlier).

In other words:

$\sqrt{30}\cdot\sqrt{1}= \\ \downarrow\\ \sqrt{30}\cdot\sqrt[2]{1}=\\ \sqrt{30}\cdot 1^{\frac{1}{2}}=\\ \sqrt{30} \cdot1=\\ \boxed{\sqrt{30}}$

Therefore, the correct answer is answer C.

Answer

$\sqrt{30}$

Exercise #4

Solve the following exercise:

$\sqrt{16}\cdot\sqrt{1}=$

Video Solution

Step-by-Step Solution

Let's start by recalling how to define a root as a power:

$\sqrt[n]{a}=a^{\frac{1}{n}}$

Next, we will remember that raising 1 to any power will always yield the result 1, even the half power of the square root.

In other words:

$\sqrt{16}\cdot\sqrt{1}= \\ \downarrow\\ \sqrt{16}\cdot\sqrt[2]{1}=\\ \sqrt{16}\cdot 1^{\frac{1}{2}}=\\ \sqrt{16} \cdot1=\\ \sqrt{16} =\\ \boxed{4}$ Therefore, the correct answer is answer D.

Answer

$4$

Exercise #5

Solve the following exercise:

$\sqrt{1}\cdot\sqrt{2}=$

Video Solution

Step-by-Step Solution

Let's start by recalling how to define a square root as a power:

$\sqrt[n]{a}=a^{\frac{1}{n}}$

Next, we remember that raising 1 to any power always gives us 1, even the half power we got from converting the square root.

In other words:

$\sqrt{1} \cdot \sqrt{2}= \\ \downarrow\\ \sqrt[2]{1}\cdot \sqrt{2}=\\ 1^{\frac{1}{2}} \cdot\sqrt{2} =\\ 1\cdot\sqrt{2}=\\ \boxed{\sqrt{2}}$ Therefore, the correct answer is answer a.

Answer

$\sqrt{2}$

Do you think you will be able to solve it?

Question 1

Solve the following exercise:

$\sqrt{25}\cdot\sqrt{4}=$

Question 2

Solve the following exercise:

$\sqrt{9}\cdot\sqrt{4}=$

Question 3

Solve the following exercise:

$\sqrt{100x^2}=$

Start practice

Square Root Rules

What is a root?

What is necessary to know about a root?

Perfect Squares:

Non-Perfect Squares:

Negative Numbers:

Square as a Power:

Roots in order of operations:

Simplifying Square Roots:

Test yourself on rules of roots!

What is a square root anyway?

Laws of Radicals

The root of a product

Square Root of a Quotient

Root of a Radical

Square Root of a product

Square Root of a quotient

Square Root of a Radical

Examples and exercises with solutions on properties of roots

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

Rules of Roots Combined