Square Root: Definition, Rules, Table, How to Find

The square root of a number is an important concept in the number system. Note that the square root is the inverse method of finding squares. In this section, we will discuss about the square root of a number. For more details, visit our page on “the properties of square roots“.

What is square root? A number multiplied by itself is called the square of that number. So $x \times x=x^2$ is the square of $x.$ Here the number $x$ is called the square root of $x^2.$

Definition of Square Root

The number $x$ is called a square root of a number $y,$ if we have the following relation:

$x \times x=y.$

Thus, if $y=x^2$ then we say $x$ is the square root of $y.$

Symbol of Square Root

Square root symbol: Symbolically, the square root is represented by $\sqrt{}.$ This symbol is also known as the radical symbol or root symbol. Recall that the number inside the root symbol $\sqrt{}$ is called the radicand. For example, $7$ is the radicand of $\sqrt{7}.$

From the above definition, we conclude that $\sqrt{x \times x}=\sqrt{x^2}=x.$ Observe that $x^2=x \times x$ is the radicand of $x=\sqrt{x \times x}.$ Hence, square is the radicand of the square root.

Square Root Examples

Few examples of square roots:

Note that $3^2=3\times 3=9.$ So by definition $\sqrt{9}=3.$

As $4^2=4\times 4=16,$ we have $\sqrt{16}=4.$

We have $5^2=5\times 5=25.$ Thus $\sqrt{25}=5.$ For more details, click on the square root of 25.

Rules of Square Root

We list all the properties of square roots.

• $x^{1/2}=\sqrt{x}$
• $\sqrt{x^2}=x$
• $a\sqrt{x}+b\sqrt{x}=(a+b)\sqrt{x}$
• $a\sqrt{x}-b\sqrt{x}=(a-b)\sqrt{x}$
• $\sqrt{x \times y}=\sqrt{x} \times \sqrt{y}$
• $\sqrt{\frac{x}{y}}=\frac{\sqrt{x}}{\sqrt{y}}$   if $y \neq 0$

Square Root Formula

The square root $\sqrt{y}$ can be expressed according to the rule of indices as follows: $\sqrt{y}=y^{1/2}.$ If the number $y$ is a perfect square, that is, $y=x \times x$ for some number $x$ then the formula to find the square root of $y$ is given below:

$\sqrt{y}=\sqrt{x \times x}=x.$

Square Root Table

Square toot table: (perfect square)

Square Root as a Function

Note that $(-x)^2=x$ and $(+x)^2=x.$ Thus we have $\sqrt{x^2}=\pm x.$ As the square root of a number takes two values, it does not make a function. But if we define $\sqrt{x^2}=+x,$ then it will definitely make a function.  Now define a function

$f: \mathbb{R} \to \mathbb{R} \,\, \text{by} \,\, f(x)=\sqrt{x}.$

Note that $f$ is a one-to-one function but not onto.

Methods of Finding Square Root

To find the square root of a number, the following methods are generally used.

• Square root by prime factorization
• Square root by repeated subtraction method
• Square root by estimation method
• Square root by long division method

Simplifying Square Roots

Now, we will learn how to simplify square roots. Let us simplify the square root of $27$.

Question: Simplify $\sqrt{27}$

Solution: Firstly, we will factorize $27.$ Note that

$27=3 \times 3 \times 3$

Taking square root on both sides, we get that

$\sqrt{27}=\sqrt{3 \times 3 \times 3}$

$=\sqrt{3 \times 3} \times \sqrt{3}$  $[\because \sqrt{x \times y}=\sqrt{x} \times \sqrt{y}]$

$=3 \times \sqrt{3}$  $[\because \sqrt{a \times a}=a]$

$=3\sqrt{3}$

$\therefore \sqrt{27}=3\sqrt{3}.$ So the square root of $27$ is $3\sqrt{3}$ (YouTube video 0f √27).

Addition or Subtraction of Square Roots

How to add/subtract two square roots? The following steps have to be followed.

Step 1: First, we express the square roots into their simplified forms as above.

Step 2: If the simplified forms contain only the same radical $\sqrt{x}$, then apply the formula $a\sqrt{x}\pm b\sqrt{x}=(a\pm b)\sqrt{x}$ to get the desired sum or difference. If they have different radicals then we keep them as it is.

For example, add $\sqrt{50}$ and $\sqrt{18}$

Question: Find $\sqrt{50}+\sqrt{18}$

Solution: Firstly, we will simplify both square roots.

$\sqrt{50}=\sqrt{25 \times 2}$ $=\sqrt{25} \times \sqrt{2}$ $=5\sqrt{2}$

$\sqrt{18}=\sqrt{9 \times 2}$ $=\sqrt{9} \times \sqrt{2}$ $=3\sqrt{2}$

So $\sqrt{50}+\sqrt{18}$ $=5\sqrt{2}+3\sqrt{2}$ $=8\sqrt{2}.$ Youtube video link of find √50+√18.

Similarly, $\sqrt{50}-\sqrt{18}$ $=5\sqrt{2}-3\sqrt{2}$ $=2\sqrt{2}$

Multiplication or Division of Square Roots

How to multiply or divide two square roots? The below steps have to be followed.

Step 1: First, we express the square roots into their simplified forms as above.

Step 2: Then applying the formula $a\sqrt{x} \times b\sqrt{y}=ab\sqrt{xy}$ we get the desired multiplication. To divide the square roots, we have to use the formula $a\sqrt{x} \div b\sqrt{y}=a/b \frac{\sqrt{x}}{\sqrt{y}}.$

Square Root of a Decimal

How to compute the square root of decimals? We need to follow the below steps to find the square root of a decimal number.

Step 1: At first, we have to express the decimal number as a fraction.

Step 2: In the next step, we use the formula $\sqrt{\frac{x}{y}}=\frac{\sqrt{x}}{\sqrt{y}}.$

Step 3: Now we will apply the method of finding the square root of real numbers to compute √x and √y.  And we put their values in the fraction obtained in step 2.

Step 4: At last, we will simplify the fraction obtained in step 3. The resultant value will be the square root of the given decimal number.

Let’s understand the above process by an example.

Question: Find $\sqrt{0.16}$

Solution:

Note that $0.16=\frac{16}{100}$

Taking square root we get

$\sqrt{0.16}=\sqrt{\frac{16}{100}}$ $=\frac{\sqrt{16}}{\sqrt{100}}$ $=\frac{4}{10}$ $=0.4$

So the square root of $0.16$ is $0.4$

Square Root of a Negative Number

We know that the square of any number is always positive. So by definition, the square root of a negative number does not exist in the set of real numbers. But they do exist in the world of complex numbers.

For example, let’s consider a negative number $x.$ So $x=-r$ for some positive number $r.$ Now $\sqrt{x}=\sqrt{-r}$ $=\sqrt{-1 \times r}$ $=\sqrt{-1} \sqrt{r}$ $=i\sqrt{r},$ where $i=\sqrt{-1}$ is a root of $-1$. The number $i$ is a purely imaginary number.

Thus we have $\sqrt{-4}=i\sqrt{4}$ $=2i.$ So the square root of $-4$ is $2i.$

Square Root of a Complex Number

The computation of the square root of a complex number is not as simple as the method of finding the square root of a real number. We know that the general form of a complex number is $a+ib,$ where both $a$ and $b$ are real numbers. The formula for finding the square root of $a+ib$ is given below:

$\sqrt{a+ib}=$ $\pm \Big[\sqrt{\frac{a+\sqrt{a^2+b^2}}{2}} + i\sqrt{\frac{-a+\sqrt{a^2+b^2}}{2}} \Big]$

Using the above formula, we can find the square root of $i.$ Note that $a=0$ and $b=1$ as $i=0+i \cdot 1.$ So we have

$\sqrt{i}=\pm (\sqrt{\frac{1}{2}}+i\sqrt{\frac{1}{2}})$ $=\pm \frac{1}{\sqrt{2}}(1+i).$

In a similar way, one can find the square root of $-i$ also.

Square Root of a 2×2 matrix

Square root of a matrix: Let $A$ and $B$ be two matrices such that $A^2=B.$ In this case, we say that the matrix $A$ is the square root of $B.$ Note that there may have many square roots of $B,$ for example, $I_2,$ the 2×2 identity matrix, has infinitely many square roots. So it is natural to ask the question: is the square root of a matrix well-defined? The answer is yes. We will discuss it now.

Let $B$ be a positive-definite matrix. We know that there is only one positive-definite matrix $A$ such that $A^2=B.$ So we define

$\sqrt{B}=A.$

The square root of $B$ is well-defined. We note that among the infinite number of square roots of $I_2,$ there is exactly one positive-definite which we define to be the square root of $I_2.$

Generalization of Square Roots

We know that the square root $x$ of the number $y$ satisfies the relation $y=x^2.$ Mathematically, we have $\sqrt{y}=x.$ This can be generalized in a broad manner.

Cube root: If $x^3=y$ then $x$ is called a cube root of $y.$ We write it as $\sqrt[3]{y}=x.$ In exponent form, the cube root can be expressed as power $1/3,$ that is, $\sqrt[3]{y}=y^{1/3}.$

Fourth root: If $x^4=y$ then $x$ is called a fourth root of $y.$ It is written as $\sqrt[4]{y}=x.$ In exponent form $\sqrt[4]{y}=y^{1/4}.$

Fifth root: If $x^5=y$ then $x$ is called a fifth root of $y.$ We write it as $\sqrt[5]{y}=x.$ In exponent form $\sqrt[5]{y}=y^{1/5}.$

n-th root: If $x^n=y$ then $x$ is called a n-th root of $y.$ We write it as $\sqrt[n]{y}=x.$ In exponent form $\sqrt[n]{y}=y^{1/n}.$

Polynomial root: Let $f(x)$ be a polynomial. If a number $c$ satisfies the equation $f(c)=0,$ then $c$ is called a root of $f(x).$ This type of roots are known as polynomial roots.

Square Root Applications

The computation of square roots has many applications in several branches of mathematics; such as

• Equation solving
• Polynomial
• Ring theory, field theory
• Numerical analysis
• Geometry (for example, to find the side of a square if the area is known)

Solved Problems of Square Roots

Question: Find the square root of $125.$

Solution:

Note that $125=5 \times 5 \times 5$

So $\sqrt{125}=\sqrt{5 \times 5 \times 5}$

$=\sqrt{5 \times 5} \times \sqrt{5}$  $[\because \sqrt{x \times y}=\sqrt{x} \times \sqrt{y}]$

$=5 \times \sqrt{5}$  $[\because \sqrt{a \times a}=a]$

$=5\sqrt{125}$

So the value of the square root of $125$ is $5\sqrt{5}.$

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