Lesson 13: Calculating and Comparing Square Roots

New Concept

The inverse of squaring a number is taking its square root. The square root of $x$, written $\sqrt{x}$, is the number whose square is $x$.

What’s next

We begin our journey with the square root, the inverse of squaring. Next, you'll tackle worked examples on finding, estimating, and comparing these important new numbers.

Property

A perfect square is a number that is the square of an integer. The product of an integer and itself is a perfect square, like $2^2 = 4$.

Examples

$8^2 = 8 \times 8 = 64$. So, 64 is a perfect square.
The number 50 is not a perfect square because no integer multiplied by itself equals 50.
$15^2 = 225$. So, 225 is a perfect square.

Explanation

Think of it as building a flawless square out of tiles! If a number of tiles can form a perfect square shape with no leftovers, that number is a perfect square. Many numbers, like 10, will leave you with extra pieces and can't do it.

Property

The square root of a number $x$, written as $\sqrt{x}$, is the number that, when squared, equals $x$. It is the inverse of squaring, so if $s^2 = A$, then $s = \sqrt{A}$.

Examples

Since $4^2 = 16$, the square root of 16 is 4, which is written as $\sqrt{16} = 4$.
A square sandbox has an area of 169 square feet. The side length is $\sqrt{169} = 17$ feet.
The solution to the equation $b = \sqrt{4}$ is $b=2$.

Explanation

Finding a square root is like being a geometry detective! If you know the total area of a square room, the square root tells you the exact length of one of its sides. It’s the ultimate “undo” button for the squaring operation you just learned.

Property

To estimate the square root of a non-perfect square, first find the two perfect squares it is between. The integer whose square is closer to your number is your estimate.

Examples

• To estimate $\sqrt{50}$, notice 50 is between $49(7^2)$ and $64(8^2)$. Since 50 is closer to 49, $\sqrt{50} \approx 7$.
• $\sqrt{37}$ is between $\sqrt{36}=6$ and $\sqrt{49}=7$. Because 37 is closer to 36, we estimate $\sqrt{37} \approx 6$.
• $\sqrt{40}$ is between the whole numbers 6 and 7, since $6^2=36$ and $7^2=49$.

Explanation

This is like being a number line detective! You trap the tricky, non-perfect square root between two “friendly” perfect squares. Whichever perfect neighbor it’s cozier with gives you the closest whole number guess for its value. It's a great trick for quick estimations!