Radicals, Exponents, and Units
Key Takeaways
- Exponent laws (product, quotient, power, zero, negative) apply to integer and rational exponents when bases are positive or when roots are defined.
- Write a^(m/n) as (n-th root of a)^m to evaluate expressions such as 16^(3/4) = 8.
- Simplify radicals by factoring out perfect squares (or cubes) and rationalizing denominators when the context requires exact form.
- Dimensional analysis multiplies by conversion factors equal to 1, canceling units until the target unit remains.
- ETS 5165 provides an on-screen calculator, but symbolic exponent setup should be done by hand before you compute.
Radicals and Exponents on the 5165 Blueprint
ETS lists radicals and rational exponents alongside units and dimensional analysis in the 30% Number & Quantity and Algebra domain. Items may look symbolic (simplify sqrt(72)) or applied (convert 55 mph to feet per second). Secondary math teachers are expected to connect notation to meaning, not just punch a calculator.
Integer Exponent Rules
For nonzero bases a and b and integers m, n:
| Rule | Statement |
|---|---|
| Product | a^m · a^n = a^(m+n) |
| Quotient | a^m / a^n = a^(m−n) |
| Power of power | (a^m)^n = a^(mn) |
| Zero exponent | a^0 = 1 (a ≠ 0) |
| Negative exponent | a^(−n) = 1/a^n |
Worked Example 1
Simplify (2x^3 y^(−2))^2 / (4x y^4).
Numerator: 4x^6 y^(−4). Divide: (4x^6 y^(−4))/(4x y^4) = x^5 y^(−8) = x^5 / y^8.
Worked Example 2 — Negative exponents
Evaluate 3^(−2) · 3^5 = 3^3 = 27. Also 2^(−4) = 1/16.
Rational Exponents
a^(m/n) = (n-th root of a)^m = n-th root of a^m (for a ≥ 0 when n is even).
Worked Example 3 — 16^(3/4)
Fourth root of 16 is 2. Cube it: 2^3 = 8.
Worked Example 4 — 27^(−2/3)
27^(1/3) = 3, so 27^(2/3) = 9. The negative exponent gives 1/9.
Worked Example 5 — Rewriting radicals
sqrt(x^5) = x^(5/2) for x ≥ 0. This form helps when multiplying radical expressions with variables.
Simplifying Radicals
Factor out perfect powers:
- sqrt(72) = sqrt(36·2) = 6 sqrt(2)
- cbrt(54) = cbrt(27·2) = 3 cbrt(2)
- sqrt(48x^3) = 4x sqrt(3x) for x ≥ 0
Rationalize denominators with radicals:
5/sqrt(3) = (5 sqrt(3))/3.
(2)/(sqrt(5) − 1) multiply by conjugate: 2(sqrt(5)+1)/4 = (sqrt(5)+1)/2.
Praxis items may ask which form is equivalent—check both coefficient and radicand.
Operations with Radicals
Like radicals add: 3 sqrt(2) + 5 sqrt(2) = 8 sqrt(2).
Unlike radicals do not: sqrt(2) + sqrt(3) cannot simplify further.
Multiply: sqrt(6) · sqrt(10) = sqrt(60) = 2 sqrt(15).
Units and Dimensional Analysis
Treat units like algebraic factors. A conversion factor such as (5280 ft / 1 mi) equals 1.
Worked Example 6 — Speed conversion
Convert 55 miles per hour to feet per second.
55 (mi/h) × (5280 ft / 1 mi) × (1 h / 3600 s) = 55 × 5280 / 3600 ≈ 80.7 ft/s.
Set up so miles cancel and hours cancel, leaving ft/s.
Worked Example 7 — Area units
A room is 4.5 m × 3.2 m. Find area in square feet (1 m ≈ 3.28 ft).
Area = 14.4 m². Because area is squared, (3.28 ft/m)² ≈ 10.76 ft²/m².
14.4 × 10.76 ≈ 155 ft² (reasonable check: ~14 m² is ~150 ft²).
Worked Example 8 — Density
Gold density 19.3 g/cm³. Convert to kg/m³.
19.3 g/cm³ × (1 kg / 1000 g) × (100 cm / 1 m)³ = 19.3 × 1000 = 19,300 kg/m³.
Common Traps
- Adding exponents only when multiplying same base—not when adding terms (2^3 + 2^4 ≠ 2^7).
- Taking sqrt(x^2) as x instead of |x| when x can be negative.
- Forgetting to square the conversion factor in area or volume problems.
- Evaluating even roots of negative numbers in reals—undefined.
Teaching Connection
When a student writes sqrt(a + b) = sqrt(a) + sqrt(b), identify the distributive error—exponent rules do not distribute over addition. Correct with a counterexample: sqrt(9 + 16) = 5, but sqrt(9) + sqrt(16) = 7. When a student converts mph to ft/s but forgets to convert hours to seconds, the answer is off by a factor of 3600—trace units in the denominator.
Scientific Notation with Exponents
Multiply (3 × 10^4)(2 × 10^7) = 6 × 10^11. Divide coefficients and subtract exponents: (8 × 10^9)/(2 × 10^3) = 4 × 10^6.
Worked Example 10
Simplify (2 × 10^−3)^2 = 4 × 10^−6 = 0.000004.
Sign Rules Under Even and Odd Roots
sqrt(x^2) = |x|. For x = −5, sqrt((−5)^2) = sqrt(25) = 5, not −5. Cube roots accept negative inputs: cbrt(−8) = −2.
When variables appear under even roots, assume nonnegative x unless the stem specifies otherwise.
Multi-Step Unit Chains
Worked Example 11 — Medication rate
A drip delivers 250 mL in 2 hours. How many milliliters per minute?
250 mL / 2 h × (1 h / 60 min) = 250/(120) ≈ 2.08 mL/min.
Chain conversions one factor at a time and verify units cancel. ETS wrong answers often match a single missed conversion factor.
Estimation for Sanity Checks
Before accepting a calculator result, estimate: sqrt(50) is a bit more than 7, so 7.07 is plausible; 70.7 is not. Estimation catches misplaced decimal points in unit problems.
Combining Radical and Exponent Forms
Worked Example 12
Write x^(2/3) · x^(1/3) = x^(3/3) = x. Exponent laws apply to rational exponents exactly as they do to integers.
When simplifying sqrt(12) + sqrt(27), rewrite as 2 sqrt(3) + 3 sqrt(3) = 5 sqrt(3)—combine only like radicals.
ETS Calculator Discipline
Use the on-screen graphing calculator for decimal checks after simplifying symbolically. Entering sqrt(72) without simplification may yield a long decimal that does not match a multiple-choice exact form.
What is the value of 16^(3/4)?
Which expression is equivalent to sqrt(50)?