x²y⁶/xy²

October 19, 2026

2022 Handbook

2022 Test A Math · Question 7 of 57

Question

$\frac{x^{2}y^{6}}{xy^{2}}$

What is the value of the expression above if $x = -2$ and $y = 2$ ?

A−64
B−32
C16
D32

Show the answer

Correct answer: B

First, substitute −2 for x and 2 for y.

$\frac{(-2)^{2}(2)^{6}}{(-2)(2)^{2}}$

Then, simplify.

$\frac{(-2)(-2)(2)^{2}(2)^{4}}{(-2)(2)^{2}}$

$\frac{(-2)(2)^{4}}{1}$

$\frac{-2 \times 16}{1}$

$-32$

Question analysis

This question asks for the value of a fraction built from powers of x and y when x is −2 and y is 2. Students evaluate an expression with exponents and a negative number, or simplify it first by cancelling common factors. Keeping track of the negative sign through the squaring and the division is the tricky part.

NYS standards: 6.EE.2c evaluating expressions and order of operations
7.NS.2a rational number multiplication
7.NS.2b rational number division
Grade level: Grade 7 content, by NYS standard
Compared with grade 7: At grade 7 Grade 7 students evaluate expressions with negative values and whole-number exponents. This one is a single fraction of powers, routine in method, but raising a negative base to powers and then dividing leaves several chances to lose the sign.
Estimated difficulty: Straightforward: two steps or one careful reading.
Thinking it demands: Depth of knowledge 1, recall and reproduction · mainly apply
What makes it hard: Error-prone arithmetic moderate
Something you must already know minor
Solution: 3 steps
Reading level: Question and choices: about grade 8, 13 words.
Students, last 30 days: Not enough answers yet to show (it takes 10).

Why it is hard, traps and wrong answers (shows the answer)

Why it is hard

Two routes work: substitute and compute 256 ÷ (−8), or simplify to xy⁴ first and evaluate (−2)(16). Simplifying first is safer, because it leaves a single negative factor to track. Substituting first brings in (−2)², where treating the minus sign as outside the parentheses flips the final sign, and each listed wrong value comes from an exponent or a sign handled wrongly.

Error-prone arithmetic moderate Working it out
Substituting first gives (−2)²(2)⁶ over (−2)(2)², so the work runs through a squared negative, 2⁶ = 64 and a negative divisor, and one factor of −2 decides the sign. (Brown & Burton, 1978; Junker & Sijtsma, 2001)
Something you must already know minor Working it out
The quotient rule for powers is not stated; students who know that x²/x = x and y⁶/y² = y⁴ can reduce the expression to xy⁴ before any number goes in. (Anderson & Krathwohl, 2001)

Strategy: Cancel the factors of x and y that the top and bottom share before substituting, so there is less to compute, and track the sign of every factor of −2 through to the end.

Traps

Order of operations: Working left to right, or distributing a minus sign wrongly, gives a choice.
A faulty procedure: A systematic but wrong procedure: adding fractions straight across, applying an exponent to each term of a sum, dropping a sign when moving a term.

The wrong answers, classified

A A faulty procedure (Brown & Burton, 1978; Tatsuoka, 1983)
Reveals a slip Expected pull: weak
Drops the square on y in the denominator, dividing 4 × 64 = 256 by (−2)(2) = −4 instead of by (−2)(4) = −8, giving −64.
C A faulty procedure (Brown & Burton, 1978; Tatsuoka, 1983)
Reveals a misconception Expected pull: moderate
Cancels x² against x as if x²/x were 1 rather than x, so only y⁶/y² = y⁴ = 16 is left, with no factor of −2.
D Order of operations (Brown & Burton, 1978)
Reveals a misconception Expected pull: strong
Squares as if the minus sat outside the parentheses, taking (−2)² as −4, so −4 × 64 = −256 and −256 ÷ (−8) = 32.

How the questions are analyzed, and the research behind it

Worked solution, step by step

Formulas used

A power: $a^{4} = a \times a \times a \times a$; a is the base, the number being multiplied. The small number is the exponent. It says how many times the base is used as a factor, a number being multiplied. So y⁶ means six y’s multiplied together. It does not mean y times 6.
Dividing powers with the same base: $\frac{a^{m}}{a^{n}} = a^{m - n}$; a is the base. m and n are exponents. The top has m factors of a and the bottom has n. Each matching pair cancels, because a ÷ a = 1. So m − n factors are left. A letter with no exponent, like x, has exponent 1.

Sort out the question
You need the value of $\frac{x^{2}y^{6}}{xy^{2}}$ when x = −2 and y = 2. The top means $x^{2} \times y^{6}$ . The bottom means $x \times y^{2}$ . You could put the numbers in right away. But simplifying first is safer. It leaves only one negative number to track.
Simplify the x part
Use the rule for dividing powers. The x on the bottom has exponent 1.
$\frac{x^{2}}{x} = x^{2 - 1} = x^{1} = x$
You can see it by writing it out: $\frac{x \times x}{x}$ leaves one x. Watch out: x² ÷ x is x, not 1. If you cancel every x, you lose the −2 and get 16, choice C.
Simplify the y part
$\frac{y^{6}}{y^{2}} = y^{6 - 2} = y^{4}$
So the whole fraction becomes a simpler expression:
$\frac{x^{2}y^{6}}{xy^{2}} = x \times y^{4}$
Watch out: the bottom has y², not y. Using y leaves y⁵ and leads to −64, choice A.
Put in the numbers
Now substitute x = −2 and y = 2.
$x \times y^{4} = (-2) \times 2^{4}$
$2^{4} = 2 \times 2 \times 2 \times 2 = 16$
$(-2) \times 16 = -32$
A negative times a positive is negative. So the answer is B, −32.

Check your answer

Substitute first, the long way. Top: $(-2)^{2} \times 2^{6} = 4 \times 64 = 256$ . Bottom: $(-2) \times 2^{2} = (-2) \times 4 = -8$ . Then $256 \div (-8) = -32$ . Watch out: (−2)² is +4, since two negatives multiply to a positive. Writing −4 leads to 32, choice D.

The original DOE question

Question 64 as printed on page 81 of the DOE handbook. — As printed in the DOE’s NYC Guide to the Specialized High Schools Admissions Test (SHSAT) for 2023 Admissions, page 81, question 64. Open page 81 of the handbook (PDF) The handbook letters this question’s choices E–H; here they are A–D (E = A, F = B, G = C, H = D).