4.1 Exponents & Square Roots

Exponent Notation: Base and Exponent

An exponent tells you how many times to use a base as a factor. In 5³, the base is 5 and the exponent is 3, so 5³ = 5 × 5 × 5 = 125. Read it as "five to the third power" or "five cubed."

The base is the number being multiplied; the small raised exponent counts how many copies of the base appear in the product.

• 2⁴ = 2 × 2 × 2 × 2 = 16 (base 2, exponent 4)
• 7² = 49 ("seven squared")
• 10³ = 1000 ("ten cubed")

When the exponent is 2, we say squared; when it is 3, cubed.

Powers of integers (3 as base)

Powers of simple fractions

Apply the exponent to the numerator and denominator separately.

• (2/3)² = 2²/3² = 4/9
• (1/5)³ = 1³/5³ = 1/125
• (3/4)² = 9/16

Sign with negative bases

A negative base with an even exponent gives a positive result; with an odd exponent it stays negative.

• (−2)³ = (−2)(−2)(−2) = −8
• (−2)⁴ = (−2)(−2)(−2)(−2) = +16

Trap: −2⁴ is not the same as (−2)⁴. Without parentheses the exponent binds first, so −2⁴ = −(2⁴) = −16. Parentheses decide whether the minus sign is part of the base.

Basic Exponent Rules

Let a be any nonzero number and let m, n be integers.

1. Product of powers: aᵐ · aⁿ = aᵐ⁺ⁿ — add the exponents when the bases match.

   • 2³ · 2⁴ = 2³⁺⁴ = 2⁷ = 128
   • x² · x⁵ = x⁷

2. Quotient of powers: aᵐ / aⁿ = aᵐ⁻ⁿ — subtract exponents.

   • 2⁷ / 2³ = 2⁷⁻³ = 2⁴ = 16
   • x⁵ / x² = x³

3. Power of a power: (aᵐ)ⁿ = aᵐⁿ — multiply the exponents.

   • (2³)² = 2³ˣ² = 2⁶ = 64
   • (x²)³ = x⁶

4. Power of a product: (ab)ᵐ = aᵐbᵐ.

   • (2x)³ = 2³ · x³ = 8x³

5. Zero exponent: a⁰ = 1 (for a ≠ 0). Any nonzero number to the zero power is 1.

   • 7⁰ = 1, 999⁰ = 1, (3/4)⁰ = 1

6. Negative exponent: a⁻ⁿ = 1/aⁿ. A negative exponent means "take the reciprocal," not "make the number negative."

   • 2⁻³ = 1/2³ = 1/8
   • 5⁻² = 1/25
   • 3⁻¹ = 1/3
   • (1/2)⁻³ = (2/1)³ = 8 ← reciprocate first, then apply the positive exponent

Perfect Squares 1² through 15²

A perfect square is an integer made by squaring a whole number. Memorize this list — NEX estimation questions lean on it.

Knowing 11² = 121, 12² = 144, 13² = 169, 14² = 196, 15² = 225 saves time on estimation problems.

Square Roots

The principal square root of a number n, written √n, is the nonnegative number that, when squared, gives n.

• √49 = 7, because 7² = 49
• √100 = 10, because 10² = 100
• √0 = 0

Square root and squaring are inverse operations, but only for nonnegative inputs:

• (√a)² = a for a ≥ 0
• √(a²) = |a| — the √ symbol always returns a nonnegative value

This second form matters with negatives: √((−5)²) = √25 = 5, not −5.

Approximating Non-Perfect Square Roots

What is √50? 50 is not a perfect square, but it sits between 49 (= 7²) and 64 (= 8²). So √50 is between 7 and 8, slightly above 7. Estimate: √50 ≈ 7.07 (check: 7.07² ≈ 49.98).

Method:

1. Find the two nearest perfect squares that bracket your number.
2. Take the root of each.
3. Estimate where your number falls between them.

Example: √30 lies between √25 = 5 and √36 = 6, a bit above 5. Estimate ≈ 5.48 (5.48² ≈ 30.03).

Simplifying Simple Radicals

A radical is in simplest form when the number under the root has no perfect-square factor larger than 1.

Simplify √72:

1. Find the largest perfect-square factor of 72. That's 36, since 36 · 2 = 72.
2. Rewrite: √72 = √(36 · 2) = √36 · √2.
3. Take the root of the perfect-square part: 6√2.

So √72 = 6√2.

Another: √45 = √(9 · 5) = √9 · √5 = 3√5.

Squaring and Square Root as Inverses

Squaring and taking a square root "undo" each other for nonnegative inputs:

• √64 = 8, then 8² = 64.
• 12² = 144, then √144 = 12.

Be careful: the equation x² = 25 has two solutions, x = 5 and x = −5, because both 5² and (−5)² equal 25. But the symbol √25 equals 5 only — the principal root is nonnegative.

Trap (sign with even powers): (−3)² = 9, but −3² = −9. The parentheses decide whether the minus sign is inside the squaring.

Trap (√ is nonnegative): √16 = 4, never ±4. The equation x² = 16 has two solutions, but the √ symbol means the nonnegative one.