Square Root Calculator for Square Root Formula, Table and Examples Learning

What is the square root of $2$? Can you answer this math quiz? If not, it's fine. Many students find square roots tricky at first, but I’m here to make them easy for you. Whether you’re stuck trying to figure out a square root question, confused by the square root symbol, or wondering what a square root calculator does, this guide will help.

Understanding square roots is like learning the secret shortcut to solving math problems quickly. Ever heard of a root mean square or wondered how to calculate the square root of a square? Don’t be intimidated by these terms. Have no fear, by the end of this article you’ll know exactly how to tackle these concepts and even score well on classwork as well as questions such as, what is square $\sqrt{64}$ ? So, read through this guide and get ready—this journey into square roots is about simplifying your math life in ways you didn’t expect!

What is a Square Root?

At its core, the square root is a number that, when multiplied by itself, gives you the original number. For example, the square root of $16$ is $4$ because $4×4=16$. Similarly, the square root of $9$ is $3$ because $3×3=9$. This simple relationship between numbers is why square roots are considered the inverse of squaring. Square roots are written using the square root symbol ($\sqrt{}$). Inside the symbol is a number (called a radicand). For example, in $\sqrt{25}$ , the radicand is $25$. When solving square root problems, you might hear about something called the root mean square or see questions about numbers like the square root of $2$. All of these terms lead to finding a number that will square back to its original value.

The Square Root Formula

You can express the square root of a number using exponents. The formula is:

$\sqrt{n} = n^{1/2}$

This formula shows that finding the square root of a number is the same as raising it to the power of $\frac{1}{2}$.

This notation is useful in various mathematical contexts, especially when dealing with exponents and algebraic manipulations. Here are a few examples to illustrate this:

1. For $n = 81$: $\sqrt{81} = 81^{1/2} = 9$.
2. For $n = 25$: $\sqrt{25} = 25^{1/2} = 5$.

This equivalence between the square root and the exponent $\frac{1}{2}$ is a fundamental concept in mathematics.

How to Find the Square Root

Finding the square root of a number might seem challenging, but with the right methods, it’s a breeze! Below I went through some of the commonly used techniques, from simple tricks to a tad more advanced numbers calculations. Whether you’re using a square root calculator or solving by hand, this guide will help you understand how to find the square root of a square.

Using a Square Root Calculator

If you’re in a hurry, a square root calculator can save the day. Take Mathos AI's Square Root Calculator as an example: you can just simply input the number, and it will instantly provide the square root. For example, type in "Find the square root of $4$" in Mathos AI:

This tool is especially useful for non-perfect squares, like finding the square root of $2$. You can ask follow-up questions in Mathos AI:

Estimation Method

This method involves guessing a number close to the square root and refining your guess:

• Start with two numbers that the square root lies between. For example, the square root of $50$ lies between $7$ and $8$ because $7 × 7 = 49$ and $8 × 8 = 64$.
• Average these numbers: $(7 + 8) ÷ 2 = 7.5$.
• Square your estimate to see how close it is: $7.5 × 7.5 = 56.25$. Adjust your guess and repeat until you’re satisfied with the result.

Example: Estimate the square root of $5$

To estimate the square root of $5$, we can use the method of successive approximations. Let's start with an initial guess and refine it.

Initial Guess:

We know that $2^2 = 4$ and $3^2 = 9$. Therefore, $\sqrt{5}$ is between $2$ and $3$. Let's start with an initial guess of $2.5$.

Refinement Using Average:

We can refine our guess using the formula:

$\text{New guess} = \frac{\text{Old guess} + \frac{5}{\text{Old guess}}}{2}$

• First iteration:

$\text{Old guess} = 2.5$

$\text{New guess} = \frac{2.5 + \frac{5}{2.5}}{2} = \frac{2.5 + 2}{2} = \frac{4.5}{2} = 2.25$

• Second iteration:

$\text{Old guess} = 2.25$

$\text{New guess} = \frac{2.25 + \frac{5}{2.25}}{2} = \frac{2.25 + 2.2222}{2} \approx \frac{4.4722}{2} \approx 2.2361$

Further Refinement:

We can continue refining, but let's check the accuracy of our current estimate:

$2.2361^2 \approx 4.9997 \approx 5$

Thus, the estimated value of $\sqrt{5}$ is approximately $2.2361$.

Prime Factorization Method

This technique works best for perfect squares:

• Break the number into its prime factors. For example, $36 = 2 × 2 × 3 × 3$.
• Pair the prime factors: $(2 × 3) × (2 × 3)$.
• Take one number from each pair: $2 × 3 = 6$. So, the square root of $36$ is $6$.

Example: Use prime factorization method to find the square root of 8

To find the square root of $8$ using the prime factorization method, follow these steps:

Prime Factorization of $8$:

$8 = 2 \times 2 \times 2 = 2^3$

Express the Square Root:

$\sqrt{8} = \sqrt{2^3}$

Simplify the Square Root:

We can rewrite $2^3$ as $2^2 \times 2$:

$\sqrt{2^3} = \sqrt{2^2 \times 2} = \sqrt{2^2} \times \sqrt{2} = 2 \sqrt{2}$

Therefore, the square root of $8$ is:

$\sqrt{8} = 2\sqrt{2}$

Long Division Method

The long division method is ideal for non-perfect squares and large numbers:

• Pair digits of the number starting from the decimal point, grouping two digits at a time.
• Find the largest number whose square is less than or equal to the first pair. Subtract and bring down the next pair of digits.
• Double the quotient as the new divisor and repeat the steps until you reach the desired precision.

Example: Use the long division method to find the square root of $69$

To find the square root of $69$ using the long division method, follow these steps:

Set up the number in pairs:

Write $69$ as $69.00$ (adding decimal places for precision).

Find the largest number whose square is less than or equal to the first pair ($69$):

The largest number whose square is less than or equal to $69$ is $8$, because $8^2 = 64$.

Subtract and bring down the next pair of digits:

Double the quotient and use it as the new divisor:

Double the current quotient ($8$) to get $16$. Write it as $160$ (since we will bring down the next pair of digits).

Find the next digit:

Find a digit $x$ such that $160x \times x$ is less than or equal to $500$. The digit $x$ is $3$ because $163 \times 3 = 489$.

Subtract and bring down the next pair of digits:

Double the current quotient ($83$) to get $166$. Write it as $1660$ (since we will bring down the next pair of digits).

Find the next digit:

Find a digit $y$ such that $1660y \times y$ is less than or equal to 1100. The digit $y$ is $0$ because $1660 \times 0 = 0$.

Continue the process for more precision:

Thus, the square root of 69 is approximately 8.30. For more precision, you can continue the process further.

Repeated Subtraction Method

For smaller, perfect square numbers, this method is simple:

• Keep subtracting consecutive odd numbers from the given number until you reach $0$.
• Count how many subtractions it took. That’s the square root! For instance, for $16$:
  • $16 - 1 = 15$
  • $15 - 3 = 12$
  • $12 - 5 = 7$
  • $7 - 7 = 0$ The square root of $16$ is $4$ because it took four steps.

Square Root Table

A quick glance at a square root table can save you time during exams. Here is a list of square roots for the numbers from $1$ to $10$:

## Questions Most Students Asked

Square Root of a Negative Number

Negative numbers don’t have real square roots because squaring any number, positive or negative, always gives a positive result. However, in advanced math, imaginary numbers solve this issue. The square root of a negative number involves the concept of imaginary numbers. The imaginary unit is denoted by $i$, where $i$ is defined as:

$i = \sqrt{-1}$

For a negative number $-a$ (where $a > 0$), the square root can be expressed as:

$\sqrt{-a} = \sqrt{a} \cdot \sqrt{-1} = \sqrt{a} \cdot i$

For example, the square root of $-9$ is:

$\sqrt{-9} = \sqrt{9} \cdot \sqrt{-1} = 3i$

Thus, the square root of a negative number always involves the imaginary unit $i$.

How to Find the Square Root of A Square Root

To find the square root of a square root, you can use the property of exponents. The square root of a number $x$ is written as $\sqrt{x}$, which is equivalent to $x^{1/2}$. Therefore, the square root of $\sqrt{x}$ can be written as:

$\sqrt{\sqrt{x}} = \sqrt{x^{1/2}}$

Using the property of exponents $(a^m)^n = a^{m \cdot n}$, we get:

$\sqrt{x^{1/2}} = (x^{1/2})^{1/2} = x^{(1/2) \cdot (1/2)} = x^{1/4}$

So, the square root of a square root of $x$is:

$\sqrt{\sqrt{x}} = x^{1/4}$

For example, if $x = 16$:

Since $16 = 2^4$, we have:

$16^{1/4} = (2^4)^{1/4} = 2^{4 \cdot (1/4)} = 2^1 = 2$

Thus, $\sqrt{\sqrt{16}} = 2$.

How to Simplify Square Roots

Simplifying a square root makes working with large numbers easier. Follow these steps:

1. Factor the number into primes.
2. Group pairs of the same factors.
3. Move one number from each pair outside the radical.

Let's go through an example of simplifying $\sqrt{72}$ to illustrate these steps:

1. Factor 72 into its prime factors:

$72 = 2 \times 36 = 2 \times 6 \times 6 = 2 \times 2 \times 3 \times 2 \times 3 = 2^3 \times 3^2$

1. Pair the prime factors:

$72 = 2^3 \times 3^2 = (2 \times 2) \times 2 \times (3 \times 3)$

1. Move each pair of prime factors outside the square root:

$\sqrt{72} = \sqrt{2^3 \times 3^2} = \sqrt{(2^2 \times 2) \times 3^2} = \sqrt{2^2} \times \sqrt{3^2} \times \sqrt{2} = 2 \times 3 \times \sqrt{2} = 6\sqrt{2}$

So, the simplified form of $\sqrt{72}$ is:

$\sqrt{72} = 6\sqrt{2}$

Square Number Required Exam Questions

Square roots often pop up in math exams, especially in questions about perfect squares or solving equations. Examples include:

Solve for $x$: $x^2 = 49$

See how Mathos AI solves this question:

Simplify: $sqrt{50}$

Knowledge about these concepts can let you solve algebra problems and quadratic equations intelligently.

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