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College Algebra Unit 1: Real numbers, exponents & radicals

Everything in algebra rests on how numbers and exponents behave. This unit sets up the rules you'll use for the rest of the course.

Covers: Number sets, exponent rules, radicals & rational exponents, scientific notation

Study guide

The real numbers are organized into nested sets: natural numbers (1, 2, 3, …), whole numbers (adding 0), integers (adding negatives), rational numbers (ratios of integers, including terminating and repeating decimals), and irrational numbers like √2 and π whose decimals never repeat. Together the rationals and irrationals make up the reals.

Exponent rules simplify products and powers: aᵐ·aⁿ = aᵐ⁺ⁿ, aᵐ/aⁿ = aᵐ⁻ⁿ, (aᵐ)ⁿ = aᵐⁿ, (ab)ⁿ = aⁿbⁿ, a⁰ = 1 and a⁻ⁿ = 1/aⁿ. A negative exponent moves a factor across the fraction bar; it doesn't make the number negative.

Radicals and rational exponents are two notations for the same thing: a^(1/n) = ⁿ√a and a^(m/n) = (ⁿ√a)ᵐ. Simplify radicals by pulling out perfect powers (√50 = 5√2), combine only like radicals, and rationalize denominators by multiplying by a radical (1/√3 = √3/3) or a conjugate for binomial denominators.

Scientific notation writes numbers as a × 10ⁿ with 1 ≤ |a| < 10. To multiply, multiply the coefficients and add the exponents; to divide, divide and subtract. Then adjust so the coefficient is back between 1 and 10.

The sum of a rational number and an irrational number is irrational, and so is a nonzero rational times an irrational. Two irrationals, however, can sum or multiply to a rational number, as with √2·√2 = 2. To compare numbers like 2⁴⁰ and 3²⁵, rewrite them with a common exponent: 2⁴⁰ = (2⁸)⁵ = 256⁵ and 3²⁵ = (3⁵)⁵ = 243⁵, so 2⁴⁰ is larger. Common bases or common roots work the same way.

An exponent applies only to what it touches: (−2)⁴ = 16, but −2⁴ = −(2⁴) = −16.

Key ideas

• Natural ⊂ whole ⊂ integers ⊂ rationals ⊂ reals; irrationals like √2 and π can't be written as fractions.
• Exponent rules work for any nonzero base and let you simplify products, quotients and powers.
• A rational exponent is a root: a^(m/n) = (ⁿ√a)^m.
• Simplify radicals by pulling out perfect-power factors; rationalize denominators to remove roots from the bottom.
• Scientific notation writes a number as a × 10ⁿ with 1 ≤ |a| < 10.

Formulas and rules

Common mistakes

• Writing (a + b)² = a² + b². The middle term 2ab is missing.
• Thinking √(a + b) = √a + √b.

Words to know

Product of powers

aᵐ·aⁿ = aᵐ⁺ⁿ

Negative exponent

a⁻ⁿ = 1/aⁿ

Rational exponent

a^(m/n) = (ⁿ√a)ᵐ: 8^(2/3) = (∛8)² = 4.

Simplifying a square root

Pull out perfect-square factors: √50 = 5√2.

Rationalizing the denominator

Removing radicals from a denominator: 1/√3 = √3/3.

Rational number

A ratio of integers; includes repeating decimals like 0.333… = 1/3.

Natural numbers

1, 2, 3, …

Integers

… −2, −1, 0, 1, 2 …

Irrational number

A real number that isn't a ratio of integers, like √2 or π.

Quotient of powers

aᵐ/aⁿ = aᵐ⁻ⁿ

Power of a power

(aᵐ)ⁿ = aᵐⁿ

Scientific notation

a × 10ⁿ with 1 ≤ |a| < 10.

Like radicals

Same index and radicand; only these can be added.

Sample practice questions

27 more questions like these, from level 1 to 5, are in the practice set.