Square Roots and Perfect Squares

🟩 Square Roots and Perfect Squares

Part 1 of 5 — Perfect Squares

Topics in This Part

🔑 Key Concept: A perfect square is what you get when you multiply a whole number by itself. Before we can find a square root, we need to know our perfect squares cold — that's Part 1.

What Is a Perfect Square?

To square a number means to multiply it by itself. We write it with a small raised $2$, called an exponent:

$5^2 = 5 \times 5 = 25$

A number is a perfect square if it equals some whole number squared. So $25$ is a perfect square because $25 = 5^2$.

The First Twelve Perfect Squares

💡 Why "square"? A square with sides of length $5$ has an area of $5 \times 5 = 25$ square units. Squaring literally measures the area of a square!

Square These Numbers 🧮

Multiply each number by itself.

1) $6^2 = \,?$ 2) $9^2 = \,?$ 3) $12^2 = \,?$

Concept Check 🎯

Spotting Non-Perfect-Squares

Not every number is a perfect square. The perfect squares are spread out: $1, 4, 9, 16, 25, 36, \dots$ Numbers between them — like $5$, $20$, or $30$ — are not perfect squares.

⚠️ Watch out: A number can look like a square but not be one. $48$ ends in $8$ and is even, but it sits between $36$ and $49$ — not a perfect square. Always ask: "Is there a whole number that, times itself, gives this?"

You now know the perfect squares up through $144$. In Part 2 we run this idea backward to find square roots.

Perfect Square or Not? 🔽

Decide whether each number is a perfect square.

🟩 Square Roots and Perfect Squares

Part 2 of 5 — Finding Square Roots

🔑 The Idea: A square root undoes squaring. Squaring asks "what is $5$ times itself?" ($25$). A square root asks the reverse: "what number, times itself, gives $25$?" ($5$).

The Radical Symbol

The square root is written with a radical sign $\sqrt{\phantom{x}}$:

$\sqrt{25} = 5 \quad\text{because}\quad 5^2 = 25$

The number under the radical is called the radicand. Reading $\sqrt{25}$ aloud: "the square root of twenty-five."

Squaring and Square-Rooting Are Inverses

$5 \xrightarrow{\;\text{square}\;} 25 \xrightarrow{\;\sqrt{\phantom{x}}\;} 5$

They cancel each other out, just like $+$ and $-$, or $\times$ and $\div$.

💡 Shortcut: To find $\sqrt{n}$ for a perfect square, just ask "which number in my times-table squared gives $n$?" The perfect squares from Part 1 are your lookup table.

Find the Square Root 🧮

Each radicand is a perfect square, so the answer is a whole number.

1) $\sqrt{16} = \,?$ 2) $\sqrt{49} = \,?$ 3) $\sqrt{121} = \,?$

Concept Check 🎯

Two Special Square Roots

Two roots are worth memorizing because they trip people up:

$\sqrt{1} = 1 \quad\text{because}\quad 1^2 = 1 \times 1 = 1$

$\sqrt{0} = 0 \quad\text{because}\quad 0^2 = 0 \times 0 = 0$

⚠️ Common slip: $\sqrt{1}$ is $1$, not $0$. And the square root of a number is usually smaller than the number itself (for example $\sqrt{36} = 6$), but for $1$ they're equal, and there's no square root smaller than $0$ in pre-algebra.

🔑 You can't square-root a negative (in pre-algebra). Since any number times itself is positive or zero, no real number squares to a negative. So $\sqrt{-9}$ has no answer here.

Match Each Root 🔽

Choose the value of each square root.

🟩 Square Roots and Perfect Squares

Part 3 of 5 — Estimating Non-Perfect Roots

🔑 The Idea: Most numbers aren't perfect squares, so their roots aren't whole numbers. We can still estimate $\sqrt{n}$ by trapping it between the two nearest perfect squares.

Trapping a Root Between Two Whole Numbers

$\sqrt{30}$ isn't a whole number, but we can pin it down. Find the perfect squares just below and above $30$:

$25 < 30 < 36 \quad\Longrightarrow\quad \sqrt{25} < \sqrt{30} < \sqrt{36} \quad\Longrightarrow\quad 5 < \sqrt{30} < 6$

So $\sqrt{30}$ is between $5$ and $6$. Because $30$ is closer to $36$ than to $25$, $\sqrt{30}$ is a bit closer to $6$ — about $5.5$.

The Estimating Recipe

1. Find the largest perfect square less than the radicand.
2. Find the smallest perfect square greater than the radicand.
3. The root lands between their two roots.

💡 Worked Example: Estimate $\sqrt{50}$. The nearest perfect squares are $49 = 7^2$ and $64 = 8^2$, so $7 < \sqrt{50} < 8$. Since $50$ is just barely above $49$, $\sqrt{50} \approx 7.1$.

Concept Check 🎯

Trap the Root 🧮

For each, enter the whole number just below the square root.

1) $\sqrt{20}$ is between $\,?$ and the next whole number. 2) $\sqrt{55}$ is between $\,?$ and the next whole number. 3) $\sqrt{99}$ is between $\,?$ and the next whole number.

Roots That Never Stop

When the radicand is a perfect square, the root is a tidy whole number. When it isn't, the root is a decimal that goes on forever without repeating — we call that irrational.

$\sqrt{2} = 1.41421356\ldots \quad(\text{never ends, never repeats})$

For now you don't need the exact value — a good estimate or leaving it as $\sqrt{2}$ is enough.

⚠️ Don't write $\sqrt{2} = 1.41$ with an equals sign — it's only approximately $1.41$. Use $\approx$ ("approximately equal").

Whole or Irrational? 🔽

Will each square root be a whole number, or an irrational decimal?

🟩 Square Roots and Perfect Squares

Part 4 of 5 — Using Square Roots

🔑 Big Payoff: Square roots aren't just abstract — they answer real questions. "If a square garden has area $64$ $ft^{2}$, how long is each side?" The answer is $\sqrt{64} = 8$ ft.

Side Length from Area

The area of a square is $A = s^2$, where $s$ is the side length. To go backward from area to side, take the square root:

$s = \sqrt{A}$

Worked Example: A Square Patio

A square patio covers $81$ square feet. How long is each side?

$s = \sqrt{81} = 9 \text{ feet}$

✅ Check: $9 \times 9 = 81$ square feet. ✓

Find the Side Length 🧮

Each square has the given area. Find the side length (a whole number of feet).

1) Area $= 36$ $ft^{2}$. Side $= \,?$ ft 2) Area $= 49$ $ft^{2}$. Side $= \,?$ ft 3) Area $= 121$ $ft^{2}$. Side $= \,?$ ft

Square Roots in the Pythagorean Theorem

For a right triangle with legs $a$ and $b$ and hypotenuse $c$:

$a^2 + b^2 = c^2 \quad\Longrightarrow\quad c = \sqrt{a^2 + b^2}$

The final step always needs a square root to undo the squaring.

Worked Example: The 3-4-5 Triangle

Legs $a = 3$ and $b = 4$:

$c = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5$

💡 The $3$-$4$-$5$ right triangle is the most famous one in math — its sides are all whole numbers because $25$ is a perfect square. Builders use it to make perfect right angles!

Concept Check 🎯

Build the Solution 🔽

A right triangle has legs $9$ and $12$. Work toward the hypotenuse.

🟩 Square Roots and Perfect Squares

Part 5 of 5 — Mixed Practice & Mastery Check

You can now (1) recognize perfect squares, (2) find square roots, (3) estimate roots of non-perfect squares, and (4) use roots in area and Pythagorean problems. Let's put it all together.

Quick Reference

⚠️ Remember: $\sqrt{0} = 0$ and $\sqrt{1} = 1$, you can't take a real square root of a negative, and roots of non-perfect-squares are irrational (use $\approx$, not $=$).

Mixed Practice 🎯

Mixed Drill 🧮

1) $\sqrt{169} = \,?$ 2) A square has area $100$ $m^{2}$. Its side length is $\,?$ m. 3) Right triangle legs $5$ and $12$: hypotenuse $= \sqrt{25 + 144} = \,?$

Exit Quiz ✅

Answer all three to finish the lesson.