Lesson 12-2: Understand and Use Square Roots

Property

The symbol $\sqrt{A}$ (square root) indicates a number $a$ whose square is $A$: $a^2 = A$. The square root $\sqrt{A}$ is only defined for non-negative numbers $A$. A positive integer whose square root is a positive integer is called a perfect square.

Examples

• Since $8^2 = 64$, the square root of 64 is 8. We write this as $\sqrt{64} = 8$. The number 64 is a perfect square.
• The number 50 is not a perfect square. Its square root, $\sqrt{50}$, is a number that, when multiplied by itself, equals 50.
• To find the number whose square is 121, we are looking for $\sqrt{121}$. Since $11 \times 11 = 121$, the answer is $11$.

Explanation

A square root is the opposite of squaring a number. If you know the area of a square, the square root tells you the side length. Perfect squares are special because their square roots are nice, neat whole numbers!

Property

The area ($A$) of a square is found by squaring its side length ($s$):

To find the side length of a square given its area, you take the square root of the area:

Examples

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!