3.2 Radicals, Rational Exponents & the Laws of Exponents

3.2 Radicals, Rational Exponents & the Laws of Exponents

Competency 1 skill 8 sits in the number sense competency rather than the algebra competency because it is about numerical radicals and rational exponents. Competency 2 skill 3 handles the same machinery applied to variable expressions. Learn the laws once here; they transfer directly.

The laws of exponents

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|  Product        a^m * a^n = a^(m+n)          2^3 * 2^4 = 2^7 = 128        |
|  Quotient       a^m / a^n = a^(m-n)          5^6 / 5^2 = 5^4 = 625        |
|  Power of power (a^m)^n = a^(mn)             (3^2)^4 = 3^8 = 6561         |
|  Power of prod  (ab)^n = a^n * b^n           (2*5)^3 = 8 * 125 = 1000     |
|  Power of quot  (a/b)^n = a^n / b^n          (3/4)^2 = 9/16               |
|  Zero exponent  a^0 = 1, a != 0              17^0 = 1                     |
|  Negative exp   a^(-n) = 1 / a^n             4^(-2) = 1/16                |
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The single most common error is adding exponents when bases differ. 2³ · 3³ is not 6⁹ and not 2⁶; the product rule requires identical bases. Here the power of a product rule applies in reverse: 2³ · 3³ = (2 · 3)³ = 6³ = 216.

The second most common error is multiplying exponents on a product of like bases. 2³ · 2⁴ = 2⁷ = 128, not 2¹² = 4096. Adding is for multiplying terms; multiplying is for a power raised to a power.

Which part is the base?

An exponent attaches only to the symbol immediately to its left unless parentheses say otherwise, and items are written to punish the confusion.

• −3² means −(3²) = −9, because the exponent attaches to the 3 and the negative sign is applied afterward.
• (−3)² = (−3)(−3) = 9, because the parentheses make −3 the base.
• −3³ = −27 and (−3)³ = −27 agree here only because an odd power preserves the sign.
• 2 · 5² = 2 · 25 = 50, not 10² = 100; the exponent does not reach the coefficient.

Even and odd roots of negative numbers

An odd index accepts a negative radicand: ∛(−27) = −3, because (−3)³ = −27. An even index does not: √(−9) is not a real number, since no real number squared is negative. This is why √(x²) equals |x| rather than x, and it is the reason a rational exponent with an even denominator, such as (−16)^(1/4), has no real value.

Rational exponents

A fractional exponent is a compact way to write a root:

a^(1/n) = ⁿ√a and a^(m/n) = ⁿ√(aᵐ) = (ⁿ√a)ᵐ

The denominator is the index of the root; the numerator is the power. Read 8^(2/3) as "the cube root of 8, squared."

Evaluate root-first whenever possible — the numbers stay small:

• 8^(2/3) = (∛8)² = 2² = 4 (versus ∛64 = 4, same answer, larger intermediate)
• 16^(3/4) = (⁴√16)³ = 2³ = 8
• 32^(2/5) = (⁵√32)² = 2² = 4
• 27^(−2/3) = 1/(∛27)² = 1/3² = 1/9
• (25/49)^(1/2) = 5/7

The negative-exponent case above is worth pausing on: a negative rational exponent produces a reciprocal, not a negative number. 27^(−2/3) = 1/9, never −9.

Simplifying radicals

To simplify √n, factor out the largest perfect square.

• √72 = √(36 · 2) = √36 · √2 = 6√2
• √200 = √(100 · 2) = 10√2
• √48 = √(16 · 3) = 4√3
• ∛54 = ∛(27 · 2) = 3∛2 (perfect cubes for a cube root)

The product and quotient rules for radicals are the engine: √(ab) = √a · √b and √(a/b) = √a / √b, both for nonnegative a and positive b.

[!WARNING] There is no sum rule for radicals. √(9 + 16) = √25 = 5, but √9 + √16 = 3 + 4 = 7. The claim √(a + b) = √a + √b is false, and it is one of the most-tested student misconceptions in middle-grades mathematics.

Combining radicals

Radicals add and subtract as like terms only when the index and radicand match.

• 3√5 + 7√5 = 10√5
• √2 + √3 — cannot be combined
• √50 + √18 = 5√2 + 3√2 = 8√2 — simplify first, then the terms match

That last example is the standard item design: two radicals that look unlike become like terms after simplification.

Multiplication does not require matching radicands: √3 · √12 = √36 = 6, and 2√5 · 3√10 = 6√50 = 6 · 5√2 = 30√2.

Rationalizing denominators

Convention keeps radicals out of denominators. Multiply by a form of 1.

Monomial denominator: multiply by the radical over itself.

6/√3 = (6/√3)(√3/√3) = 6√3/3 = 2√3

Binomial denominator: multiply by the conjugate, which changes the sign between the terms. The product of conjugates is a difference of squares, eliminating the radical.

4/(3 + √5) = 4(3 − √5)/((3 + √5)(3 − √5)) = 4(3 − √5)/(9 − 5) = 4(3 − √5)/4 = 3 − √5

Where this shows up

Rational exponents surface in compound-growth arithmetic: a quantity that grows by a factor of 8 over 3 years grows by 8^(1/3) = 2 per year. They surface in geometry: a cube of volume 64 cubic units has edge 64^(1/3) = 4, and a square of area 50 has side √50 = 5√2 ≈ 7.07. And they surface in scaling: if area scales by a factor of 9, linear dimensions scale by 9^(1/2) = 3.