Pre-Calc Diagnostic: Simplify Radicals & Exponents (Stewart Problem 2)

Direct answers:

(a) 200 − 32 = 62

(b) (3a³b³)(4ab²)² = 48a⁵b⁷

(c) (3x^3/2y³x²y^−1/2)−2 = x9y⁷

Below, each one is worked the way it should be reasoned through on the Stewart diagnostic — and why these exact patterns become reflexes that AP Calculus, the SAT, and college algebra all rely on.

Why Problem 2 exists

If Problem 1 tests whether a student knows the exponent rules in isolation, Problem 2 tests whether they can combine them. Real pre-calc and AP Calc problems never give you a single rule to apply — they give you an expression that needs two or three rules in sequence, and they punish students who can't keep them straight.

The three expressions here are designed to be cumulative. Part (a) is one rule. Part (b) is two rules. Part (c) is essentially every exponent rule you know, applied in the right order. Let's walk through them.

(a) 200 − 32 — radical simplification

You can't subtract 200 and 32 directly because they don't share the same radicand. The whole game in radical simplification is finding the largest perfect-square factor inside each radical so that what's left under the root is the same.

Step 1. Factor each radicand using perfect squares:

200 = 100 · 2, so 200 = 100 · 2 = 100 · 2 = 102

32 = 16 · 2, so 32 = 16 · 2 = 16 · 2 = 42

Step 2. Both radicals now share 2, so subtract the coefficients:

102 − 42 = 62

The trick is recognizing perfect-square factors on the first try. Memorize the perfect squares up to 400 cold: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, 256, 289, 324, 361, 400. Students who hesitate on "what's the largest perfect square that divides 200?" lose time on every radical problem.

(b) (3a³b³)(4ab²)² — power of a product, then product rule

Two rules, in this order: distribute the outer exponent first, then combine like bases.

Step 1: power-of-a-product. The outer squared parentheses apply to every factor inside:

(4ab²)² = 4² · a² · (b²)² = 16 · a² · b⁴ = 16a²b⁴

Notice how (b²)² uses the power-of-a-power rule: multiply the exponents, 2 · 2 = 4.

Step 2: multiply, then use the product rule. The original expression is now (3a³b³)(16a²b⁴). Multiply the numerical coefficients and add the exponents on like bases:

• 3 · 16 = 48
• a³ · a² = a^{3 + 2} = a⁵
• b³ · b⁴ = b^{3 + 4} = b⁷

Putting it together: 48a⁵b⁷.

The lesson: always distribute the outer exponent first, before trying to combine anything. Students who try to multiply (3a³b³)(4ab²) first and then square the result almost always make a sign or coefficient error.

(c) (3x^3/2y³x²y^−1/2)−2 — the boss expression

This one looks brutal. It is not. It's the same three rules applied carefully, in order. The students who get this wrong almost always make the same mistake: they try to simplify what's inside the parentheses first, then apply the outer −2. Wrong order. Distribute the outer exponent first; the inside cleans up automatically.

Step 1: distribute the outer −2 to every factor using power-of-a-quotient. The −2 applies to each piece in the numerator and denominator:

(3x^3/2y³x²y^−1/2)−2 = 3⁻² · x^(3/2)(−2) · y^(3)(−2)x^(2)(−2) · y^{(−1/2)(−2)}

= 3⁻² · x⁻³ · y⁻⁶x⁻⁴ · y¹

Step 2: rewrite 3⁻² using the negative-exponent rule: 3⁻² = 19. Pull the 19 out front.

= 19 · x⁻³ · y⁻⁶x⁻⁴ · y¹

Step 3: apply the quotient rule on each like base — subtract the denominator exponents from the numerator exponents:

• x: x⁻³x⁻⁴ = x^{−3 − (−4)} = x^{−3 + 4} = x¹ = x
• y: y⁻⁶y¹ = y^{−6 − 1} = y⁻⁷

So now we have 19 · x · y⁻⁷.

Step 4: the instruction said "without negative exponents." Move y⁻⁷ to the denominator (negative exponent = reciprocal):

= 19 · x · 1y⁷ = x9y⁷

Five rules used: power-of-a-quotient, power-of-a-power (built into step 1), negative exponent (twice), product rule (implicit when combining 19 and x), and the quotient rule. Every one of these is a single-line rule from Problem 1's five-rule list. The only new skill in part (c) is keeping them straight in order.

The order rule that fixes 90% of student errors on (c)

Here is the single most important habit we teach SOMATH pre-calc students for multi-rule expressions like part (c):

1. Distribute the outermost exponent first (using power-of-a-product or power-of-a-quotient).
2. Then handle negative exponents (flip them).
3. Then combine like bases (product rule or quotient rule).
4. Then clean up (rewrite fractional exponents as radicals if the problem asks, or vice versa).

Out-of-order work is the single biggest source of lost points on these. A student who tries to simplify 3x^3/2y³x²y^−1/2 before applying the −2 will end up with the wrong sign on every exponent. Distribute first, simplify second.

What this problem reveals about Stewart's diagnostic

Stewart wrote Problem 2 to force three skills simultaneously: radical fluency (part a), basic exponent combination (part b), and multi-rule discipline (part c). Each part is independent in difficulty, but they're a unit pedagogically. A student who breezes through (a) and (b) but gets stuck on (c) needs more practice with rule sequencing. A student who hesitates on (a) needs to drill perfect squares before anything else.

The diagnostic isn't a test — it's a map. It tells you exactly which sub-skills are missing before AP Calculus assumes them.

How we teach this at SOMATH

If a student walks into our Upper West Side classroom at 226 W 79th St with weak radical or exponent fluency, here's the sequence we use in their first eight sessions:

1. Drill the perfect squares 1–400. Five minutes of flashcards per session until they're reflexes. No student should hesitate on "largest perfect-square factor of 72."
2. Mixed radical simplification. Twenty problems like part (a) in one sitting, with sums and differences only. The pattern locks in when the student does it twenty times in a row, not when they do one a week.
3. Two-rule exponent problems before three-rule. Like part (b) — always power-of-a-product or power-of-a-power combined with product rule. Once that's automatic, layer in negative exponents.
4. Multi-rule problems with the "distribute first" mantra. Like part (c). We have students literally annotate each step with the rule they're using — "power-of-a-quotient," "negative exponent," "quotient rule" — until the order becomes muscle memory.

By session eight, a student who started shaky on radicals can do problems like (c) in under four minutes, without notes. That's the gate AP Calc walks through in September.

FAQ

How do you simplify 200 − 32?

Pull out the largest perfect-square factor from each radical. 200 = 100 · 2 = 102, and 32 = 16 · 2 = 42. Both share 2, so subtract: 102 − 42 = 62.

How do you simplify (3a³b³)(4ab²)²?

Power-of-a-product first: (4ab²)² = 16a²b⁴. Then product rule on like bases: (3a³b³)(16a²b⁴) = 48a⁵b⁷.

How do you simplify (3x^3/2y³x²y^−1/2)−2 without negative exponents?

Distribute the −2 to every factor first. After simplification with the quotient rule and rewriting all negative exponents as reciprocals, the answer is x9y⁷.

What's the best way to memorize the exponent rules?

Write your own one-page sheet of the five rules — product, quotient, power-of-a-power, negative exponent, rational exponent. Then drill mixed-rule problems daily. We have SOMATH pre-calc students do 10–15 mixed problems a week until the rules become reflexes.

Do you offer pre-calculus tutoring on the Upper West Side?

Yes — pre-calculus and AP Calc AB/BC are core to our program. We tutor them year-round, and most of our AP Calc students take pre-calc with us the year before. Our entire program operates from 226 W 79th St on the Upper West Side.

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