What is a root?

  • A root is the inverse operation of a power.
  • It is denoted with the symbol √ √ and is equal to a power of 0.5 0.5 .
  • If a small number appears on the left, it will be the order of the root.
Start practice

Test yourself on rules of roots!

einstein

Solve the following exercise:

\( \sqrt{\frac{2}{4}}= \)

Practice more now

What is necessary to know about a root?

  1. The result of the square root will always be positive! You will never get a negative result. We can get a result of 0 0 .
  2. For \sqrt{} of a negative-number there is no answer!
  3. The square root is basically a half power. We can say that: a=a12 \sqrt{a}=a^{\frac{1}{2}}
  4. A square root precedes the four arithmetic operations. First, perform the square root and only then order the operations of the account.
The result of the square root will always 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.


Join Over 30,000 Students Excelling in Math!
Endless Practice, Expert Guidance - Elevate Your Math Skills Today
Test your knowledge

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:
(aβ‹…b)=aβ‹…b\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:
ab=ab\sqrt{\frac{a}{b}}=\frac{\sqrt{a}}{\sqrt{b}}


Do you know what the answer is?

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:
amn=anβ‹…m\sqrt[n]{\sqrt[m]{a}}=\sqrt[n\cdot m]{a}


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 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 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:

  1. The result of a root will always be positive!! A negative result will never be obtained. We can get a result of 0 0 .
  2. For √ √ (negative number) there is no answer!
  3. The root is basically a half power. We can say that: a=a12\sqrt a=a^{ 1 \over 2}
  4. A root precedes the four arithmetic operations. First, perform the root and only then order the arithmetic operations.

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

Let's ask, what power of 22 will give us 6464 and the answer is 88.
True, also βˆ’8-8 to the power of 22 will give us 6464 but the result of the root must be positive!


Check your understanding

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:
(aβ‹…b)=aβ‹…b\sqrt{(a\cdot b)}=\sqrt{a}\cdot\sqrt{b}


Let's see this in an example:
(64β‹…100)\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:
64β‹…100=\sqrt{64}\cdot\sqrt{100}=
8β‹…10=80 8\cdot10=80


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:ab=ab\sqrt{\frac{a}{b}}=\frac{\sqrt{a}}{\sqrt{b}}

Let's see this in an example:

369 \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:

369\frac{\sqrt{36}}{\sqrt{9}}

63=2\frac{6}{3}=2


Do you think you will be able to solve it?

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:
amn=anβ‹…m\sqrt[n]{\sqrt[m]{a}}=\sqrt[n\cdot m]{a}

Let's see this in the example:Β 
10042=1002β‹…4=1008\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

Solve the following exercise:

16β‹…1= \sqrt{16}\cdot\sqrt{1}=

Video Solution

Step-by-Step Solution

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

an=a1n \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:

16β‹…1=↓16β‹…12=16β‹…112=16β‹…1=16=4 \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 4

Exercise #2

Solve the following exercise:

1β‹…2= \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:

an=a1n \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:

1β‹…2=↓12β‹…2=112β‹…2=1β‹…2=2 \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

2 \sqrt{2}

Exercise #3

Solve the following exercise:

2β‹…5= \sqrt{2}\cdot\sqrt{5}=

Video Solution

Step-by-Step Solution

In order to simplify the given expression we use two laws of exponents:

A. Defining the root as an exponent:

an=a1n \sqrt[n]{a}=a^{\frac{1}{n}} B. The law of exponents for dividing powers with the same bases (in the opposite direction):

xnβ‹…yn=(xβ‹…y)n x^n\cdot y^n =(x\cdot y)^n

Let's start by changing the square roots to exponents using the law of exponents shown in A:

2β‹…5=↓212β‹…512= \sqrt{2}\cdot\sqrt{5}= \\ \downarrow\\ 2^{\frac{1}{2}}\cdot5^{\frac{1}{2}}= We continue: since we are multiplying two terms with equal exponents we can use the law of exponents shown in B and combine them together as the same base raised to the same power:

212β‹…512=(2β‹…5)12=1012=10 2^{\frac{1}{2}}\cdot5^{\frac{1}{2}}= \\ (2\cdot5)^{\frac{1}{2}}=\\ 10^{\frac{1}{2}}=\\ \boxed{\sqrt{10}} In the last steps wemultiplied the bases and then used the definition of the root as an exponent shown earlier in A (in the opposite direction) to return to the root notation.

Therefore, the correct answer is answer B.

Answer

10 \sqrt{10}

Exercise #4

Solve the following exercise:

9β‹…3= \sqrt{9}\cdot\sqrt{3}=

Video Solution

Step-by-Step Solution

Although the square root of 9 is known (3) , in order to get a single expression we will use the laws of parentheses:

So- in order to simplify the given expression we will use two exponents laws:

A. Defining the root as a an exponent:

an=a1n \sqrt[n]{a}=a^{\frac{1}{n}} B. Multiplying different bases with the same power (in the opposite direction):

xnβ‹…yn=(xβ‹…y)n x^n\cdot y^n =(x\cdot y)^n

Let's start by changing the square root into an exponent using the law shown in A:

9β‹…3=↓912β‹…312= \sqrt{9}\cdot\sqrt{3}= \\ \downarrow\\ 9^{\frac{1}{2}}\cdot3^{\frac{1}{2}}= Since a multiplication is performed between two bases with the same exponent we can use the law shown in B.

912β‹…312=(9β‹…3)12=2712=27 9^{\frac{1}{2}}\cdot3^{\frac{1}{2}}= \\ (9\cdot3)^{\frac{1}{2}}=\\ 27^{\frac{1}{2}}=\\ \boxed{\sqrt{27}} In the last steps we performed the multiplication, and then used the law of defining the root as an exponent shown earlier in A (in the opposite direction) in order to return to the root notation.

Therefore, the correct answer is answer C.

Answer

27 \sqrt{27}

Exercise #5

Solve the following exercise:

10β‹…3= \sqrt{10}\cdot\sqrt{3}=

Video Solution

Step-by-Step Solution

To simplify the given expression, we use two laws of exponents:

A. Defining the root as an exponent:

an=a1n \sqrt[n]{a}=a^{\frac{1}{n}} B. The law of exponents for dividing powers with the same base (in the opposite direction):

xnβ‹…yn=(xβ‹…y)n x^n\cdot y^n =(x\cdot y)^n

Let's start by using the law of exponents shown in A:

10β‹…3=↓1012β‹…312= \sqrt{10}\cdot\sqrt{3}= \\ \downarrow\\ 10^{\frac{1}{2}}\cdot3^{\frac{1}{2}}= We continue, since we have a multiplication between two terms with equal exponents, we can use the law of exponents shown in B and combine them under the same base which is raised to the same exponent:

1012β‹…312=(10β‹…3)12=3012=30 10^{\frac{1}{2}}\cdot3^{\frac{1}{2}}= \\ (10\cdot3)^{\frac{1}{2}}=\\ 30^{\frac{1}{2}}=\\ \boxed{\sqrt{30}} In the last steps, we performed the multiplication of the bases and used the definition of the root as an exponent shown earlier in A (in the opposite direction) to return to the root notation.

Therefore, the correct answer is B.

Answer

30 \sqrt{30}

Test your knowledge
Start practice