Square Roots and Perfect Squares

What number times itself equals 25? Understanding squares and square roots helps you work with area, the Pythagorean theorem, and many algebra problems!

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What Is a Square?

Squaring a number means multiplying it by itself.

Notation: n² = n × n

Examples:

• 3² = 3 × 3 = 9
• 5² = 5 × 5 = 25
• 10² = 10 × 10 = 100

Read as: "three squared" or "three to the second power"

Why "squared"? If you make a square with side length n, the area is n².

Example: Square with side 4 → Area = 4² = 16 square units

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Perfect Squares

Perfect squares are numbers that result from squaring whole numbers.

List of perfect squares:

• 1² = 1
• 2² = 4
• 3² = 9
• 4² = 16
• 5² = 25
• 6² = 36
• 7² = 49
• 8² = 64
• 9² = 81
• 10² = 100
• 11² = 121
• 12² = 144
• 13² = 169
• 14² = 196
• 15² = 225

Memorize at least 1² through 12²!

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What Is a Square Root?

Square root is the INVERSE (opposite) of squaring.

Question it answers: "What number, when squared, gives me this?"

Symbol: √ (radical symbol)

Definition: √n is the number that, when squared, equals n

Example: √25 = 5 because 5² = 25

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Finding Square Roots

For perfect squares:

√1 = 1 (because 1² = 1) √4 = 2 (because 2² = 4) √9 = 3 (because 3² = 9) √16 = 4 (because 4² = 16) √25 = 5 (because 5² = 25) √36 = 6 (because 6² = 36) √49 = 7 (because 7² = 49) √64 = 8 (because 8² = 8) √81 = 9 (because 9² = 81) √100 = 10 (because 10² = 100)

Pattern: √(n²) = n

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Squares and Square Roots Are Opposites

Think of them as inverse operations:

Square: Start with 5 → 5² → 25 Square root: Start with 25 → √25 → 5

They undo each other:

• √(n²) = n
• (√n)² = n

Example:

• √(7²) = √49 = 7
• (√16)² = 4² = 16

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Non-Perfect Squares

What about √20?

20 is NOT a perfect square. √20 is between √16 = 4 and √25 = 5

So √20 ≈ 4.47...

For non-perfect squares:

• Answer is NOT a whole number
• Often left in radical form: √20
• Or approximated: √20 ≈ 4.47
• Is an irrational number (decimal never ends or repeats)

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Estimating Square Roots

To estimate √50:

Step 1: Find perfect squares it's between 49 < 50 < 64 √49 < √50 < √64 7 < √50 < 8

Step 2: See which it's closer to 50 is close to 49 So √50 is a little more than 7

Estimate: √50 ≈ 7.1 (actual: 7.07...)

Example 2: Estimate √30

25 < 30 < 36 5 < √30 < 6

30 is between 25 and 36, closer to 25 Estimate: √30 ≈ 5.5 (actual: 5.48...)

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Simplifying Square Roots

Goal: Find any perfect square factors

Example 1: Simplify √20

Step 1: Factor 20 20 = 4 × 5

Step 2: Take out perfect squares √20 = √(4 × 5) = √4 × √5 = 2√5

Answer: √20 = 2√5

Example 2: Simplify √48

48 = 16 × 3 √48 = √16 × √3 = 4√3

Answer: √48 = 4√3

Strategy: Look for largest perfect square factor!

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Perfect Square Factors

Common perfect squares to look for:

• 4, 9, 16, 25, 36, 49, 64, 81, 100

Example: √72

Try factors:

• 72 = 4 × 18 → √72 = 2√18 (can simplify more!)
• 18 = 9 × 2 → √18 = 3√2
• So √72 = 2 × 3√2 = 6√2

Better: Find largest perfect square

• 72 = 36 × 2
• √72 = √36 × √2 = 6√2 (done in one step!)

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Square Roots in Equations

Example: Solve x² = 49

Take square root of both sides: x = √49 x = ±7

Wait, why ±?

Both 7² = 49 AND (-7)² = 49!

So x = 7 or x = -7

Written: x = ±7 (read as "plus or minus 7")

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Negative Numbers and Square Roots

Can you square root a negative?

In pre-algebra: NO!

Why? No real number squared gives a negative.

• Positive × Positive = Positive
• Negative × Negative = Positive

So √(-25) has no real answer!

(In advanced math, you learn about "imaginary numbers," but not yet!)

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Square Roots and Area

Finding side from area:

Problem: A square has area 144 square inches. Find the side length.

Solution: Area = side² 144 = s² s = √144 s = 12 inches

Answer: Each side is 12 inches

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Real-World Applications

Construction:

• Square room with area 400 sq ft
• Side length = √400 = 20 feet

Pythagorean Theorem:

• Right triangle: a² + b² = c²
• If a = 3, b = 4, then c² = 9 + 16 = 25
• c = √25 = 5

Physics:

• Distance fallen: d = 16t² (feet, where t is seconds)
• If d = 144 ft, then 144 = 16t², so t² = 9, t = 3 seconds

Geometry:

• Diagonal of square with side s: d = s√2
• If side = 10, diagonal = 10√2 ≈ 14.14

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Order of Operations with Radicals

PEMDAS still applies!

Remember: √ is like division (in P for Parentheses/grouping)

Example: 2 + √16 = 2 + 4 = 6

Example: √(9 + 16) = √25 = 5

Note: √9 + √16 ≠ √(9 + 16)

• √9 + √16 = 3 + 4 = 7
• √(9 + 16) = √25 = 5

Rule: Do what's inside the radical first!

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Using a Calculator

To find square roots:

• Look for √ button
• Some calculators: 2nd function + x²

Examples:

• √64 → Enter √64 = 8
• √50 → Enter √50 ≈ 7.071...

For non-perfect squares, calculator gives decimal approximation

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Common Mistakes to Avoid

❌ Mistake 1: Forgetting ± in equations

• Wrong: x² = 16, so x = 4 only
• Right: x² = 16, so x = ±4

❌ Mistake 2: Adding radicals incorrectly

• Wrong: √4 + √9 = √13
• Right: √4 + √9 = 2 + 3 = 5

❌ Mistake 3: Confusing square and square root

• 5² = 25 (multiply)
• √25 = 5 (find original number)

❌ Mistake 4: Not simplifying radicals

• Leaving √20 instead of 2√5
• Not finding perfect square factors

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Problem-Solving Strategy

Finding square roots:

1. Check if it's a perfect square (memorize list!)
2. If yes, find the whole number
3. If no, estimate or leave in radical form

Simplifying radicals:

1. Factor the number
2. Look for perfect square factors
3. Pull out perfect squares
4. Simplify

Solving equations:

1. Isolate x²
2. Take square root of both sides
3. Remember ±

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Quick Reference

Perfect Squares (memorize!): 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144

Square Root Properties:

• √(a × b) = √a × √b
• √(a²) = a (for positive a)
• (√a)² = a

Key Concepts:

• Square: n² = n × n
• Square root: inverse of squaring
• Perfect squares: result from squaring whole numbers
• ±: equations have two solutions

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Practice Tips

Tip 1: Memorize perfect squares 1-144

• Makes everything faster!
• Recognize them instantly

Tip 2: Look for patterns

• Units digit can hint at square
• Numbers ending in 2, 3, 7, 8 are never perfect squares

Tip 3: Estimate before calculating

• Helps catch errors
• Builds number sense

Tip 4: Simplify radicals completely

• Find largest perfect square factor
• Check your answer by squaring back

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Summary

Squares:

• n² means n × n
• Creates perfect squares (1, 4, 9, 16, 25...)
• Used for area of squares

Square roots:

• Inverse operation of squaring
• √n asks "what squared equals n?"
• Symbol: √

Perfect squares:

• Results from squaring whole numbers
• Memorize at least 1² through 12²
• Square roots of perfect squares are whole numbers

Non-perfect squares:

• Square roots are irrational
• Estimate or leave in radical form
• Simplify by factoring out perfect squares

Mastering squares and square roots is essential for algebra, geometry, and many real-world applications!