If we are asked to find the square root of a value, we are being asked, "What positive number multiplied by itself would give this value?" It is also the same as asking "What positive number, when squared, would give this value?"

The notation we use is the square root symbol, \sqrt{}, with a number inside it. For example, the square root of 25 is written as \sqrt{25}. Square root expressions can also be called radical expressions.

Whole number perfect squares are special numbers because their square roots are whole numbers. The sign in front of a square root determines whether the result is positive or negative. For example,\sqrt{9}=3\\ -\sqrt{9}=-3

What if we are asked to find the square root of a fraction or a decimal?

For fractions, we can find the square root of the numerator over the square root of the denominator. This is known as the division property of radicals.

\displaystyle \sqrt{\dfrac{a}{b}}

\displaystyle =

\displaystyle \dfrac{\sqrt{a}}{\sqrt{b}}

Division property of radicals

Exploration

Consider the following numbers: 0.3,\,0.06,\,1.2

Square each of the numbers and state the results.
Compare the number of decimal places in the original number to the number of decimal places in its square.
Do you think the number of decimal places in a perfect square can ever be odd? Explain your reasoning.
Compare the non-zero digits in the original number to the non-zero digits in the square.
How can we use these relationships to determine the square root of decimal numbers?

For decimals, we can use the structure of the decimal to evaluate the square root, or we can convert the decimal into a fraction before finding the square root of the numerator and the denominator.

Examples

Example 1

Evaluate \sqrt{256}.

Worked Solution

Create a strategy

We need to find a positive number that equals 256 when multiplied by itself. We can start from a perfect square that we have memorized, such as 12^2=144, then square numbers larger than 12.

Apply the idea

256 is larger than 144, which is 12^2, so we can begin by squaring 13,\,14,\,15,\, etc. until we find the solution.

\displaystyle 13\times13	\displaystyle =	\displaystyle 169
\displaystyle 14\times14	\displaystyle =	\displaystyle 196
\displaystyle 15\times 15	\displaystyle =	\displaystyle 225
\displaystyle 16 \times 16	\displaystyle =	\displaystyle 256

The answer is 16.

Example 2

Find -\sqrt{0.04}.

Worked Solution

Create a strategy

Convert the decimal into a fraction, then find the square root of the numerator over the square root of the denominator.

Apply the idea

\displaystyle -\sqrt{0.04}	\displaystyle =	\displaystyle -\sqrt{\dfrac{4}{100}}	Convert the decimal to a fraction
	\displaystyle =	\displaystyle -\dfrac{\sqrt{4}}{\sqrt{100}}	Division property of square roots
	\displaystyle =	\displaystyle -\dfrac{2}{10}	Evaluate the square roots
	\displaystyle =	\displaystyle -0.2	Convert the fraction to a decimal

The negative square root of 0.04 is -0.2.

Reflect and check

Alternatively, we could have looked at the structure of the decimal to find the square root.

4 is the only digit in the number that is not 0, and the square root of 4 is 2. There are 2 decimal places in 0.04, so there will be 2\div 2=1 decimal place in the square root.

Therefore, -\sqrt{0.04}=-0.2.

Idea summary

Finding the square root of a perfect square is looking for a number that, when multiplied to itself, will give the value.

For fractions, we can get the square root of the numerator over the square root of the denominator.

For decimals, we can:

use the structure of the decimal to evaluate the square root.
convert the decimal into a fraction.

Solving equations with square roots

Square rooting a number is the opposite of squaring a number. In other words, if a number has been squared, we can "undo" the square by taking the square root. This will help us solve equations in the form x^2=a where a is a positive, rational number.

We should remember that the square of a positive number is positive, and the square of a negative number is also positive. For example, \begin{aligned}\left(2\right)^2=4\\\left(-2\right)^2=4\end{aligned}When taking the square root of both sides of an equation, we should remember the variable could be a positive or a negative number. This is known as the square root property.

\displaystyle x^2	\displaystyle =	\displaystyle a	Original equation
\displaystyle \sqrt{x^2}	\displaystyle =	\displaystyle \pm \sqrt{a}	Square root property

Another thing to notice is that taking the square root of both sides introduces the \pm symbol. This is read as "plus or minus" and shows that there is a positive and a negative solution to the equation.

Examples

Example 3

What number, when squared, gives 25?

Worked Solution

Create a strategy

We can translate this into an equation and solve algebraically.

Apply the idea

"What number" can be represented by x, "when squared" means we need to square x, and "gives 25" means "equals 25".

\displaystyle x^2	\displaystyle =	\displaystyle 25	Translate from words to an equation
\displaystyle \sqrt{x^2}	\displaystyle =	\displaystyle \pm\sqrt{25}	Square root property
\displaystyle x	\displaystyle =	\displaystyle \pm\sqrt{25}	Evaluate the left-hand side
\displaystyle x	\displaystyle =	\displaystyle \pm 5	Evlaute the right-hand side

Reflect and check

Since this was a simple perfect square, we could have thought of the numbers without needing to solve algebraically. However, it can be easy to forget to state both the positive and negative solutions to the equation.

Using the square root property when solving algebraically can help us avoid this mistake.

Example 4

Find the solutions of x^2=\dfrac{9}{64}.

Worked Solution

Create a strategy

The equation is asking us to find numbers that, when squared, are equal to \dfrac{9}{64}. We can find the solutions by using the square root property.

Note that the square root of a fraction is the square root of the numerator over the square root of the denominator.

Apply the idea

\displaystyle x^2	\displaystyle =	\displaystyle \dfrac{9}{64}	Original equation
\displaystyle \sqrt{x^2}	\displaystyle =	\displaystyle \pm\sqrt{\dfrac{9}{64}}	Square root both sides
\displaystyle x	\displaystyle =	\displaystyle \pm\sqrt{\dfrac{9}{64}}	Evaluate \sqrt{x^2}=x
\displaystyle x	\displaystyle =	\displaystyle \pm\dfrac{\sqrt{9}}{\sqrt{64}}	Division property of square roots
\displaystyle x	\displaystyle =	\displaystyle \pm\dfrac{3}{8}	Evaluate the square roots

The solutions are x=-\dfrac{3}{8} and x=\dfrac{3}{8}.

Reflect and check

Multiplying -\dfrac{3}{8} by itself:

-\dfrac{3}{8} \times -\dfrac{3}{8}= \dfrac{9}{64}

Multiplying \dfrac{3}{8} by itself:

\dfrac{3}{8} \times \dfrac{3}{8}= \dfrac{9}{64}Therefore, our solutions are correct.

Idea summary

The square root property helps us solve equations in the form x^2=a where a is a positive, rational number.

\displaystyle x^2	\displaystyle =	\displaystyle a	Original equation
\displaystyle \sqrt{x^2}	\displaystyle =	\displaystyle \pm \sqrt{a}	Square root property

The solution may be positive or negative because the square of a positive number is positive, and the square of a negative number is also positive.

Outcomes

8.EE.A.2

Use square root and cube root symbols to represent solutions to equations of the form x^2 = p and x^3 = p, where p is a positive rational number. Evaluate square roots of small perfect squares and cube roots of small perfect cubes. Know that √2 is irrational.

1.03 Square roots of perfect squares

Introduction

Ideas

Square roots of perfect squares

Exploration

Examples

Example 1

Example 2

Solving equations with square roots

Examples

Example 3

Example 4

Outcomes

8.EE.A.2

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