Math 151 - Exponent and Radical Properties

What is the Product Property?

aᵐ · aⁿ = aᵐ⁺ⁿ — When multiplying powers with the same nonzero base, ADD the exponents.

What is the Quotient Property?

aᵐ / aⁿ = aᵐ⁻ⁿ — When dividing powers with the same nonzero base, SUBTRACT the exponents.

What is the Power of a Power Property?

(aᵐ)ⁿ = aᵐⁿ — When raising a power to another power, MULTIPLY the exponents.

What is the Power of a Product Property?

(ab)ᵐ = aᵐbᵐ — Apply the exponent to EACH factor in the product.

What is the Power of a Quotient Property?

(a/b)ᵐ = aᵐ/bᵐ — Apply the exponent to BOTH the numerator and denominator.

What is the Zero Exponent Property?

a⁰ = 1, where a ≠ 0 — Any nonzero number raised to the zero power equals 1.

What is the Negative Exponent Property?

a⁻ⁿ = 1/aⁿ, where a ≠ 0 — A negative exponent means take the reciprocal and change the exponent to positive.

What is the Rational Exponent Property?

a¹⁄ⁿ = ⁿ√a — The denominator tells you the ROOT or INDEX.

What is the General Rational Exponent Property?

aᵐ⁄ⁿ = ⁿ√(aᵐ) = (ⁿ√a)ᵐ — The denominator tells you the ROOT/INDEX; the numerator tells you the POWER.

What is the Negative Rational Exponent Property?

a⁻ᵐ⁄ⁿ = 1/aᵐ⁄ⁿ — A negative exponent means take the reciprocal and make the exponent positive.

What is the Even/Odd Exponent Rule?

For a negative base, an EVEN exponent produces a positive result; an ODD exponent produces a negative result.

Why do parentheses matter with negative numbers and exponents?

(−a)ⁿ means the exponent applies to the entire negative base. Without parentheses, the exponent applies only to a.

What is a radical expression?

An expression that contains a radical symbol, such as ⁿ√a.

What is the radicand?

The number or expression INSIDE the radical symbol.

What is the index?

The small number that indicates which ROOT to take. If no index is shown, the index is 2.

What is the index of a square root?

The index is 2, but the 2 is usually NOT written.

What does it mean to evaluate a radical?

Find the value that, when raised to the power indicated by the index, gives the radicand.

How do you convert a radical to a rational exponent?

ⁿ√a = a¹⁄ⁿ

How do you convert a rational exponent to a radical?

aᵐ⁄ⁿ = ⁿ√(aᵐ) = (ⁿ√a)ᵐ

What does the numerator of a rational exponent tell you?

The NUMERATOR tells you the POWER.

What does the denominator of a rational exponent tell you?

The DENOMINATOR tells you the ROOT or INDEX.

What is Radical Property 1?

(ⁿ√a)ⁿ = a — A matching root and exponent undo each other.

What is Radical Property 2?

ⁿ√(aⁿ) = |a| when n is even; ⁿ√(aⁿ) = a when n is odd.

Why does an even root give an absolute value?

Because an even root represents the principal (nonnegative) root, so ⁿ√(aⁿ) = |a| when n is even.

Why can an odd root keep a negative sign?

Because odd roots can produce negative values, so ⁿ√(aⁿ) = a when n is odd.

What is the Radical Product Property?

ⁿ√a · ⁿ√b = ⁿ√(ab) — Radicals with the same index can be multiplied by multiplying their radicands.

What is the Radical Quotient Property?

ⁿ√a / ⁿ√b = ⁿ√(a/b) — Radicals with the same index can be divided by dividing their radicands.