Question 1
For x > 0, the expression (x^6)^5 · x^2 is equivalent to x^n. What is the value of n?
Enter your own answer.
Answer: 32
A power of a power multiplies exponents: (x^6)^5 = x^30. Then x^30 · x^2 = x^32, so n = 32.
SAT · Math · Advanced Math
Equivalent expressions
Equivalent expressions is part of Advanced Math in the SAT Math section. The questions below are written in the style of the real test, each with the answer and a worked explanation.
Question types you will see here include expand and combine like terms, factor a quadratic expression, simplify a rational expression and radicals and rational exponents.
Common traps: sign errors when distributing negatives; missing the middle term in (a + b)²; combining unlike terms.
Sample questions
Question 2
4x² - 20x + 25 = (ax - b)² In the equation above, a and b are positive constants and the equation is true for all values of x. What is the value of ab?
- A.
2
- B.
5
- C.
7
- D.
10
Answer: D. 10
Expand: (ax - b)² = a²x² - 2abx + b². Matching coefficients, a² = 4 and b² = 25, so a = 2 and b = 5 (both positive), and -2ab = -20 confirms ab = 10. A and B are a and b alone; C is their sum rather than their product.
- A.
Question 3
Which of the following is equivalent to x² + 8x + 3, an expression a marine biologist uses to model a tide pool's water level?
- A.
(x + 8)² − 61
- B.
(x + 4)² − 13
- C.
(x + 4)² + 19
- D.
(x − 4)² − 13
Answer: B. (x + 4)² − 13
Half of 8 is 4, and (x + 4)² = x² + 8x + 16, so x² + 8x + 3 = (x + 4)² − 13 (B). A halves nothing, using (x + 8)²; C adds the balancing 16 instead of subtracting it; D uses the wrong sign inside the square, which would give −8x.
- A.
Question 4
The expression (3x - 2)(4x - 2) is equivalent to ax² + bx + c, where a, b, and c are constants. What is the value of b?
- A.
-14
- B.
-4
- C.
2
- D.
12
Answer: A. -14
Expand: (3x)(4x) + (3x)(-2) + (-2)(4x) + (-2)(-2) = 12x² + (-6 + -8)x + 4. So a = 12, b = -14, c = 4; the value of b is -14.
- A.
Question 5
A model railway is built at a fixed scale, and the area of a sign on it is 4x² − 81y² square millimetres. Which of the following is equivalent to 4x² − 81y²?
- A.
(2x − 9y)²
- B.
(2x − 9y)(2x + 9y)
- C.
(4x − 9y)(x + 9y)
- D.
(2x − 81y)(2x + y)
Answer: B. (2x − 9y)(2x + 9y)
This is a difference of squares, a² − b² = (a − b)(a + b), with a = 2x and b = 9y: 4x² − 81y² = (2x − 9y)(2x + 9y) (B). A squares a binomial, which has a middle term. C multiplies out to a middle term. D has cross terms that do not cancel.
- A.
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