Squares and Square Roots

Squaring a number means multiplying it by itself. Taking a square root reverses that process — it asks, “What number was multiplied by itself to produce this value?” These two operations are inverses, just like addition and subtraction or multiplication and division. Square roots appear in the Pythagorean theorem, distance formulas, area calculations, and electrical formulas.

What Is a Perfect Square?

A perfect square is the result of multiplying a whole number by itself. Here are the perfect squares you should memorize:

Knowing these by heart will make algebra, geometry, and standardized tests much faster.

The Square Root

The square root of a number $a$ is the value $b$ such that $b^2 = a$. We write it using the radical symbol:

$\sqrt{a} = b \quad \text{means} \quad b^2 = a$

Example 1: Evaluate $\sqrt{49}$

We need a number that, when squared, gives 49.

$7 \times 7 = 49 \quad \Rightarrow \quad \sqrt{49} = 7$

Answer: $\sqrt{49} = 7$

Example 2: Evaluate $\sqrt{144}$

$12 \times 12 = 144 \quad \Rightarrow \quad \sqrt{144} = 12$

Answer: $\sqrt{144} = 12$

Example 3: Evaluate $\sqrt{1}$ and $\sqrt{0}$

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

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

Squaring and Square Roots Are Inverses

Squaring and taking the square root undo each other:

$\sqrt{n^2} = n \quad \text{(for } n \geq 0\text{)}$

$(\sqrt{n})^2 = n \quad \text{(for } n \geq 0\text{)}$

Example 4: Verify the Inverse Relationship

Start with 8.

$8^2 = 64 \qquad \sqrt{64} = 8$

Start with 64.

$\sqrt{64} = 8 \qquad 8^2 = 64$

Either direction brings you back to where you started.

Estimating Non-Perfect Square Roots

Not every number is a perfect square. What is $\sqrt{50}$? Since 50 falls between the perfect squares 49 and 64, we know:

$\sqrt{49} = 7 \quad \text{and} \quad \sqrt{64} = 8$

So $\sqrt{50}$ is between 7 and 8. Since 50 is much closer to 49 than to 64, the answer is just a little more than 7.

Estimation method:

Step 1: Find the two consecutive perfect squares that surround the number. For 50: $49$ and $64$.

Step 2: Determine the square roots of those: $7$ and $8$.

Step 3: Figure out where the number falls between the two perfect squares.

$\frac{50 - 49}{64 - 49} = \frac{1}{15} \approx 0.07$

Step 4: Add that fraction to the lower root.

$\sqrt{50} \approx 7 + 0.07 = 7.07$

Calculator check: $\sqrt{50} \approx 7.071$. Our estimate is very close.

Example 5: Estimate $\sqrt{30}$

Step 1: The surrounding perfect squares are 25 and 36.

Step 2: $\sqrt{25} = 5$ and $\sqrt{36} = 6$.

Step 3: Where does 30 fall?

$\frac{30 - 25}{36 - 25} = \frac{5}{11} \approx 0.45$

Step 4: Estimate.

$\sqrt{30} \approx 5 + 0.45 = 5.45$

Calculator check: $\sqrt{30} \approx 5.477$. Close enough for most practical purposes.

Example 6: Estimate $\sqrt{200}$

Step 1: $196$ and $225$ are the surrounding perfect squares.

Step 2: $\sqrt{196} = 14$ and $\sqrt{225} = 15$.

Step 3:

$\frac{200 - 196}{225 - 196} = \frac{4}{29} \approx 0.14$

Step 4:

$\sqrt{200} \approx 14 + 0.14 = 14.14$

Calculator check: $\sqrt{200} \approx 14.142$. Excellent estimate.

Using a Calculator for Square Roots

On most calculators:

• Scientific calculator: Press the $\sqrt{\phantom{x}}$ button, then enter the number.
• Phone calculator: Many phone calculators have a square root button when turned to landscape/scientific mode.
• Computer: Type sqrt(number) in a search engine or use a programming language.

Calculators give decimal approximations for non-perfect squares. For exact answers on tests, leave the radical: $\sqrt{50}$ rather than $7.071$.

Real-World Application: Carpentry — Diagonal of a Square Room

A carpenter is installing a diagonal brace across a square room that measures 12 feet on each side. The diagonal is found using the Pythagorean theorem:

$d = \sqrt{12^2 + 12^2} = \sqrt{144 + 144} = \sqrt{288}$

Estimate: $\sqrt{289} = 17$, so $\sqrt{288} \approx 16.97$.

Answer: The diagonal is approximately 16.97 feet, so the carpenter should cut the brace just under 17 feet.

Real-World Application: Electrician — Current from Power

An electrician uses the formula $P = I^2 R$ (power equals current squared times resistance). If a circuit has 100 watts of power and 4 ohms of resistance, the current is:

$I^2 = \frac{P}{R} = \frac{100}{4} = 25$

$I = \sqrt{25} = 5 \text{ amps}$

Answer: The current is 5 amps.

Common Mistakes to Avoid

1. Thinking $\sqrt{a + b} = \sqrt{a} + \sqrt{b}$. This is false. For example, $\sqrt{9 + 16} = \sqrt{25} = 5$, but $\sqrt{9} + \sqrt{16} = 3 + 4 = 7$. Square roots do not distribute over addition.

2. Confusing squaring with doubling. $5^2 = 25$, not $10$. Squaring means multiplying by itself, not by 2.

3. Forgetting that square roots of negative numbers are not real. In pre-algebra, $\sqrt{-4}$ has no real answer (imaginary numbers come later in algebra 2).

4. Not simplifying when possible. $\sqrt{50}$ can be simplified to $5\sqrt{2}$ in algebra, but in pre-algebra it is fine to use the decimal approximation.

Practice Problems

Key Takeaways

• A perfect square is the result of a whole number multiplied by itself. Memorize the perfect squares from $1^2 = 1$ through $15^2 = 225$.
• The square root undoes squaring: $\sqrt{a^2} = a$ (for non-negative $a$).
• To estimate a non-perfect square root, find the two surrounding perfect squares and interpolate.
• Square roots do not distribute over addition: $\sqrt{a + b} \neq \sqrt{a} + \sqrt{b}$.
• Square roots of negative numbers have no real value at the pre-algebra level.

Return to Pre-Algebra for more topics in this section.