Math 10C · Numbers, roots and powers
Prime factors, and what an exponent means
Prime factorization is the fingerprint of a number, and it hands you the greatest common factor, the lowest common multiple and any root it contains. Exponents then extend from counting copies to fractions and negatives.
- 1. The words, first
- 2. What prime factorization gives you
- 3. Exponents, extended
- 4. Radicals and irrational numbers
- 5. What costs marks
The words, first
The idea: Most of the difficulty here is vocabulary that sounds similar and means opposite things.
| Word | What it means |
|---|---|
| Prime number | A whole number greater than 1 with exactly two factors, itself and 1. Note that 1 is not prime. |
| Prime factorization | A number written as a product of primes only: 60 = 2² × 3 × 5. |
| Greatest common factor (GCF) | The largest number dividing two or more numbers. From the factorizations, take the lowest power of each shared prime. |
| Lowest common multiple (LCM) | The smallest number that both divide into. Take the highest power of every prime that appears. |
| Perfect square / cube | A number that is a whole number squared or cubed: 49 and 125. |
| Rational number | One that can be written as a fraction of integers. Its decimal either terminates or repeats. |
| Irrational number | One that cannot. Its decimal never ends and never repeats: √2, π. |
| Radical | A root sign. The index says which root; no index means a square root. |
| Entire / mixed radical | √50 and 5√2 — the same number, written with everything inside or with the square factor pulled out. |
What prime factorization gives you
The idea: Split a number into primes once and three other questions answer themselves.
60 = 2 × 30 = 2 × 2 × 15 = 2² × 3 × 5, and 36 = 2² × 3².
GCF of 60 and 36: take the lowest power of each shared prime — 2² and 3¹ — giving 12.
LCM of 60 and 36: take the highest power of every prime present — 2², 3², 5 — giving 180.
Roots: √3600 = √(2⁴ × 3² × 5²) = 2² × 3 × 5 = 60, because a square root halves every exponent. If any exponent is odd, that prime is stuck under the root — which is how you can tell a root is irrational before evaluating it.
Useful check: GCF × LCM = the product of the two numbers. Here 12 × 180 = 2160 = 60 × 36 ✓.
Exponents, extended
The idea: Start from counting copies, then insist the same rules keep working. Negative and fractional exponents are what you get when you do.
xᵃ · xᵇ = xᵃ⁺ᵇ · xᵃ / xᵇ = xᵃ⁻ᵇ · (xᵃ)ᵇ = xᵃᵇ · (xy)ᵃ = xᵃyᵃ
Where x⁰ = 1 comes from. x³/x³ is obviously 1, and the division rule says it is x⁰. Both must be true, so x⁰ = 1.
Where a negative exponent comes from. x²/x⁵ is 1/x³ by cancelling, and x⁻³ by the rule. So x⁻³ = 1/x³ — a negative exponent means a reciprocal, not a negative answer. 3⁻² is 1/9.
Where a fractional exponent comes from. (x^½)² = x¹ = x by the power rule, and the thing that gives x when squared is √x. So x^½ = √x, and the denominator of the fraction names the root: x^(2/3) = (³√x)².
Worked: (2x³y)⁴ = 2⁴ · x¹² · y⁴ = 16x¹²y⁴. The outside exponent applies to every factor inside, and forgetting the coefficient is the usual slip.
Radicals and irrational numbers
The idea: √2 has no exact decimal, so leaving it as √2 is not laziness — it is the only exact answer available.
Simplifying. Pull out the largest perfect square: √48 = √(16 × 3) = 4√3. Going the other way, 4√3 = √(16 × 3) = √48.
Which roots are irrational? Take the prime factorization. √3600 = 60 exactly because every exponent is even. √48 = √(2⁴ · 3) leaves a 3 stuck under the root, so it is irrational.
A fact worth knowing. Rational plus irrational is always irrational — if the sum were rational, subtracting the rational part would make the irrational number rational, which is a contradiction. But irrational plus irrational can be rational: √2 + (−√2) = 0.
What costs marks
The idea: The exponent rules are where most of it happens.
- Treating a negative exponent as a negative answer. It means a reciprocal.
- Forgetting the coefficient when raising a product to a power.
- Adding exponents when multiplying bases. 2³ × 3³ is not 6⁶ — it is 6³.
- Swapping GCF and LCM. Lowest powers for the factor, highest for the multiple.