Finding a Perfect Square Root

Evaluate $\sqrt{49}$

Ask: What number times itself equals 49?: $? \times ? = 49$ = Looking for the square root

Think of perfect squares: $7 \times 7 = 49$ = 7 is the answer

Verify the answer: $7^2 = 49$ and $\sqrt{49} = 7$ = Confirmed!

Answer: $\sqrt{49} = 7$

Finding the Radicand

If $\sqrt{x} = 8$, what is $x$?

Understand what the equation means: Some number $x$, when you take its square root, equals 8 = $\sqrt{x} = 8$

Square both sides to undo the square root: $(\sqrt{x})^2 = 8^2$ = $x = 64$

Verify: take the square root of 64: $\sqrt{64} = 8$ because $8 \times 8 = 64$ = Correct!

Answer: $x = 64$

Simplifying a Radical

Simplify $\sqrt{36}$

Recognize 36 as a perfect square: $36 = 6 \times 6 = 6^2$ = 36 is a perfect square

Apply the square root: $\sqrt{36} = \sqrt{6^2}$ = Ready to simplify

The square root undoes the square: $\sqrt{6^2} = 6$ = $\sqrt{36} = 6$

Answer: $\sqrt{36} = 6$

Estimating a Non-Perfect Square Root

Estimate $\sqrt{20}$ between two integers

Find perfect squares near 20: $16 = 4^2$ and $25 = 5^2$ = 16 < 20 < 25

Take square roots of the inequality: $\sqrt{16} < \sqrt{20} < \sqrt{25}$ = $4 < \sqrt{20} < 5$

Determine which integer is closer: 20 is closer to 16 than to 25, so $\sqrt{20}$ is closer to 4 = $\sqrt{20} \approx 4.5$

Answer: $\sqrt{20}$ is between 4 and 5, closer to 4. (Exact: $\approx 4.47$)

Mistake: Thinking $\sqrt{9 + 16} = \sqrt{9} + \sqrt{16}$

Why: You cannot split a square root over addition! $\sqrt{9 + 16} = \sqrt{25} = 5$, but $\sqrt{9} + \sqrt{16} = 3 + 4 = 7$. These are different!

Correct: Always simplify inside the radical first: $\sqrt{9 + 16} = \sqrt{25} = 5$

Mistake: Forgetting that $\sqrt{x^2} = |x|$, not just $x$

Why: The square root always gives a non-negative result. For example, $\sqrt{(-3)^2} = \sqrt{9} = 3$, not $-3$.

Correct: Remember: $\sqrt{x^2} = |x|$ (absolute value)

Mistake: Confusing $\sqrt{16}$ with $16 \div 2$

Why: Square root is NOT division by 2! $\sqrt{16} = 4$ (what times itself equals 16), but $16 \div 2 = 8$.

Correct: Square root asks: 'What number squared gives me this?'

TV Screen Size

TV screens are measured diagonally. To find the diagonal, you use the Pythagorean theorem with radicals.

A TV has width 48 inches and height 36 inches. The diagonal is $\sqrt{48^2 + 36^2} = \sqrt{2304 + 1296} = \sqrt{3600} = 60$ inches.

Walking Distance

When you walk diagonally across a rectangular park, you can use radicals to find the shorter path.

A park is 300 meters by 400 meters. Walking diagonally: $\sqrt{300^2 + 400^2} = \sqrt{90000 + 160000} = \sqrt{250000} = 500$ meters.

Q: Can negative numbers have square roots?

A: In the real number system, negative numbers do not have real square roots because no real number times itself gives a negative. However, in advanced math, we use imaginary numbers (like $i = \sqrt{-1}$) to handle this.

Q: What's the difference between $\sqrt{16}$ and $\pm\sqrt{16}$?

A: $\sqrt{16} = 4$ (just the positive root). But when solving $x^2 = 16$, the answer is $x = \pm 4$ because both $4^2 = 16$ and $(-4)^2 = 16$.

Q: Why are they called 'perfect squares'?

A: Numbers like 1, 4, 9, 16, 25 are called perfect squares because they can form a perfect square shape. For example, 9 = 3 rows of 3 dots, forming a square.