7.1 Simplifying Radical Expressions
Key Takeaways
- AAF weights Radical and rational equations at 5–15% of the 20-item CAT, typically 1–3 questions covering creating, simplifying, domain, range, and graphing.
- For a,b ≥ 0, √(ab)=√a·√b, but √(a+b) is not √a+√b: √(9+16)=5, not 7.
- 4/(√7−√3) rationalizes by the conjugate √7+√3 to √7+√3 after the 4 cancels.
- For x ≥ 0 and x ≠ 4, (√x+2)/(√x−2)=(x+4√x+4)/(x−4).
- x^{1/2}=√x and x^{m/n}=(ⁿ√x)^m; 8^{2/3}=4. Skills Insight 200–236 applies the exponent rules; 250–262 rewrites rational expressions.
7.1 Simplifying Radical Expressions
College Board’s Next-Generation ACCUPLACER Advanced Algebra and Functions (AAF) test weights Radical and rational equations at 5–15% of the 20-item computer-adaptive test — typically 1–3 items. Table 11 names four skills: creating radical and rational equations and functions; domain and range; graphing; and simplifying radical and rational expressions and equations. This section is the algebra of radicals: the product and quotient rules, like radicals, rationalizing denominators (including conjugates), and fractional exponents.
Product and quotient rules
For real a ≥ 0 and b ≥ 0 (and, for the quotient, b > 0):
| Rule | Formula | Example |
|---|---|---|
| Product | √(ab) = √a · √b | √(9·5) = 3√5 |
| Quotient | √(a/b) = √a / √b | √(16/25) = 4/5 |
| Inverse of squaring | (√a)^2 = a (and `√(a^2) = | a |
The product rule is not √(a+b) = √a + √b. That false split is one of the highest-yield traps on this slice: √(9+16) = √25 = 5, while 3 + 4 = 7. Never distribute a radical across addition or subtraction.
To simplify a square root, factor out the largest perfect square. √72 = √(36·2) = 6√2. √50 = √(25·2) = 5√2. √8 = √(4·2) = 2√2. Cube roots use perfect cubes: ∛54 = ∛(27·2) = 3∛2. AAF square-root items dominate this band; cube roots appear when a fractional exponent has denominator 3.
The product rule also runs right to left: √8 · √2 = √16 = 4. Computing √8 = 2√2 first gives (2√2)(√2) = 2·2 = 4, the same answer. Prefer the direction that produces a perfect square quickly.
Indexes must match. √3 · ∛3 is not √9 and not ∛9. Convert to exponents first: 3^{1/2} · 3^{1/3} = 3^{5/6}. That conversion is why fractional exponents belong in the same lesson as radicals.
Combining like radicals
Like radicals have the same index and the same radicand. They combine the way like terms combine: 3√5 + 2√5 = 5√5. Coefficients add; the radical stays.
Unlike radicands do not combine until you simplify. √50 + √8 looks unlike, but 5√2 + 2√2 = 7√2. If the simplified radicands still differ, stop: √12 + √18 = 2√3 + 3√2 is fully simplified and does not collapse further.
| Expression | Simplified form |
|---|---|
4√7 − √7 | 3√7 |
√12 + √27 | 2√3 + 3√3 = 5√3 |
√8 − √18 + √32 | 2√2 − 3√2 + 4√2 = 3√2 |
√3 + √12 | √3 + 2√3 = 3√3 |
√45 − √20 | 3√5 − 2√5 = √5 |
A coefficient of 1 is still a like term: √5 + 6√5 = 7√5, not 6√10 and not √30. Multiplication is a different operation: (√5)(6√5) = 6·5 = 30. Adding and multiplying use the same symbols in the stem, so read the operation before you reach for a perfect-square factor.
Rationalizing a monomial denominator
A simplified radical expression does not leave an irrational number in the denominator. Multiply by a clever form of 1 that cancels the radical downstairs.
5 / √3 = 5√3 / 3.
1 / √8 should be simplified first: √8 = 2√2, so 1/(2√2) · √2/√2 = √2 / 4. Rationalizing without that first step, 1/√8 · √8/√8 = √8 / 8 = 2√2 / 8 = √2 / 4, agrees. Options often include both √8/8 and √2/4; AAF wants the fully simplified form.
The on-screen calculator (College Board calculator policy) will not rewrite 1/√3 as √3/3 for you. Handheld calculators are not allowed except with an approved accommodation. Exact form is the usual AAF target on this skill.
Conjugates: binomial denominators
If the denominator is √a − √b or √a + √b, multiply by the conjugate — the same pair with the opposite middle sign. The product is a difference of squares: (√a − √b)(√a + √b) = a − b, which is rational when a and b are rational (and not equal).
Worked example: 4 / (√7 − √3)
Multiply numerator and denominator by √7 + √3:
4 / (√7 − √3) · (√7 + √3)/(√7 + √3) = 4(√7 + √3) / (7 − 3) = 4(√7 + √3) / 4 = √7 + √3.
The 4 cancels. The fully simplified value is √7 + √3, not 4(√7 + √3). Forgetting to divide by 4 is the standard trap. A second trap uses the same sign instead of the conjugate, which produces 7 + 2√21 + 3 in the denominator instead of the rational 4. A third trap subtracts in the wrong order and writes 3 − 7 = −4, then reports −(√7 + √3).
Worked example: (√x + 2) / (√x − 2)
Assume x ≥ 0 (even root) and x ≠ 4 (the denominator is zero when √x = 2). Multiply by the conjugate of the denominator, which happens to match the numerator:
Numerator: (√x + 2)^2 = x + 4√x + 4.
Denominator: (√x − 2)(√x + 2) = x − 4.
So (√x + 2)/(√x − 2) = (x + 4√x + 4)/(x − 4) for x ≠ 4.
You can leave the numerator as (√x + 2)^2 if an option does. Do not cancel √x terms across a plus sign, and do not replace the denominator with x − 2 (that would square only one term). The exclusion x ≠ 4 survives in the simplified form because the original expression was undefined there.
Pattern, not a memorized sample: why conjugates exist
Any time a denominator is a difference of square roots, the conjugate is a difference of squares. In particular, if a numerator happens to be the matching difference of squares x − y = (√x − √y)(√x + √y), the factor √x − √y cancels when √x ≠ √y and the expression collapses to √x + √y. That identity is why conjugates earn a whole paragraph on AAF. The two worked items above use a numerical fraction 4/(√7 − √3) and an algebraic fraction (√x + 2)/(√x − 2) so you practice the method rather than a single memorized College Board sample.
Fractional exponents
| Exponent | Radical form | Example |
|---|---|---|
x^{1/2} | √x | 9^{1/2} = 3 |
x^{1/n} | ⁿ√x | 8^{1/3} = 2 |
x^{m/n} | (ⁿ√x)^m or ⁿ√(x^m) | 8^{2/3} = (∛8)^2 = 4 |
x^{-1/2} | 1/√x | 9^{-1/2} = 1/3 |
The denominator of the exponent is the index; the numerator is the power. Compute the root first when the number is a perfect power: 16^{3/4} = (⁴√16)^3 = 2^3 = 8. Computing 16^3 first is legal but painful, and it is how students land on an option such as 4096 that still needs a fourth root.
Negative exponents still mean reciprocal: 27^{-2/3} = 1 / (∛27)^2 = 1/9.
Exponent rules still apply. x^{1/2} · x^{1/3} = x^{5/6}. (x^{1/2})^3 = x^{3/2} = √(x^3) = x√x for x ≥ 0. These rewrites connect to the 250–262 band’s rational-expression work: x / √x = x^1 / x^{1/2} = x^{1/2} = √x for x > 0. Rewriting √x / x the same way gives x^{-1/2} = 1/√x.
A frequent trap treats x^{2/3} as x^2 / x^3 = 1/x. The fraction is a single exponent, not a quotient of two powers of x unless you use x^{2/3} = x^2 / x^{4/3} or another exponent identity on purpose.
Creating a radical expression from a verbal rule
Table 11 includes creating radical equations and functions, not only simplifying them. If a side length is the square root of an area A, write s = √A with domain A ≥ 0. If s = √(2x + 3) describes a length, then 2x + 3 ≥ 0 and the output is nonnegative. Skills Insight 237–249 may only ask you to solve a simple radical equation such as √(2x + 3) = 5 after checking that the inside is nonnegative — the next section, Solving Radical Equations and Extraneous Roots, solves those equations. Here, creating means writing f(x) = √(2x + 3) rather than leaving a sentence, then simplifying or evaluating.
Evaluate to check a simplification. If you claim √72 = 6√2, then at a calculator-icon item both should be near 8.49; on a no-calculator item, compare (6√2)^2 = 36·2 = 72. The square-and-compare check is faster than a decimal and matches AAF’s exact-form options.
Calculator and exact form
Some items show an on-screen calculator icon at the top right. Radical-simplification items usually want 3√2, not 4.24. If one option is √8 and another is 2√2, choose the simplified form. If both a decimal and an exact conjugate appear, the exact form matches the algebra this section teaches. Domain language for the graphs of these simplified formulas is collected in Graphs, Domain, and Range of Radical and Rational Functions.
Which expression is equivalent to 4/(√7 − √3) after rationalizing the denominator?
For x ≥ 0 and x ≠ 4, which expression is equivalent to (√x + 2)/(√x − 2)?
Which of the following is equal to x^{1/2} · x^{1/3} for x > 0?