5x^3+3x+9+1x^2 If x=10 , what is the value of the expression above?

If x = 10, what is 5x³ + 3x + 9 + 1/x²

2018 Handbook

2018 Test B Math · Question 34 of 57

Also in 2025 Practice Test B Math, question 34.

Question

$5x^{3}+3x+9+\frac{1}{x^{2}}$

If $x=10$ , what is the value of the expression above?

A2,539.01
B5,039.01
C5,039.1
D5,139

Show the answer

Correct answer: B

$5x^3+3x+9+\frac{1}{x^2}$

$=5(10^3)+3(10)+9+\frac{1}{10^2}$

$=5{,}000+30+9+\frac{1}{100}=5{,}039.01$

Question analysis

Students substitute 10 for x in an expression with a cube, a linear term, a constant and a fraction with x squared in its denominator, then add the results. It tests evaluating expressions with exponents and decimals. The fraction term is where most slips happen, because its place value is easy to get wrong.

NYS standards: 6.EE.2c evaluating expressions and order of operations
6.EE.1 whole-number exponent expressions
Grade level: Grade 6 content, by NYS standard
Compared with grade 7: Below grade 7 Evaluating expressions with whole-number exponents is grade 6 work. The only extra care here is a reciprocal term that turns into a decimal, a single routine substitution that typical grade 7 students handle comfortably.
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
A very close wrong answer moderate
Solution: 3 steps
Reading level: Question and choices: about grade 7, 11 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

Substituting 10 is easy; the risk lies in the two unlike terms at the ends. The cube must be applied to x before multiplying by 5, and the reciprocal of 100 must become a hundredth, not a tenth. With choices that differ by a single decimal place, the question rewards writing each term as a number before adding.

Error-prone arithmetic moderate Working it out
The last term, 1/x² at x = 10, is 1/100, which must be written as 0.01 and lined up beside 5,039; one place too far gives 0.1. (Brown & Burton, 1978; Junker & Sijtsma, 2001)
A very close wrong answer moderate Choosing the answer
5,039.1 sits beside the right 5,039.01 and differs only in the hundredths place; only writing 1/100 as 0.01, not 0.1, tells them apart, and estimating cannot. (Embretson & Wetzel, 1987; Gorin & Embretson, 2006; Ascalon et al., 2007)

Strategy: Work term by term: compute 10 cubed, then the reciprocal of 10 squared as a decimal, and line up the place values carefully before adding.

Traps

Order of operations: Working left to right, or distributing a minus sign wrongly, gives a choice.

The wrong answers, classified

A Order of operations (Brown & Burton, 1978)
Reveals a misconception Expected pull: weak
Gets 2,500 for the first term, most likely from (5 × 10)², applying the power to the product 5x and squaring instead of cubing.
C An arithmetic slip (Junker & Sijtsma, 2001)
Reveals a slip Expected pull: strong
Writes 1/100 as 0.1 instead of 0.01, a place-value slip on the last term that puts the total off by 0.09.
D Misunderstands the concept (Smith, diSessa & Roschelle, 1993; De Bock et al., 1998; Ni & Zhou, 2005)
Reveals a misconception Expected pull: moderate
Evaluates 1/x² as x² = 100 instead of its reciprocal, so it adds 100 rather than 0.01 to 5,039.

How the questions are analyzed, and the research behind it

Worked solution, step by step

Formulas used

Exponents: $x^3 = x \times x \times x$; x is the base number. The small raised 3 is the exponent. It tells how many times to use x in a multiplication. In the same way, x squared, written with a raised 2, means x times x.

Sort out the question
You need the value of $5x^3+3x+9+\frac{1}{x^2}$ when $x = 10$ . Put 10 in place of every $x$ . Then work out each term, one at a time. A term is a piece joined to the others by a plus sign.
Work out the first term
Do the exponent first. Then multiply by 5.
$10^3 = 10 \times 10 \times 10 = 1{,}000$
$5 \times 1{,}000 = 5{,}000$
Watch out: the exponent belongs only to the $x$ , not to the 5. Squaring $5 \times 10$ gives 2,500, which leads to choice A.
Work out the middle terms
The next term means 3 times $x$ .
$3 \times 10 = 30$
The 9 has no $x$ in it, so it stays 9.
Work out the last term
First find the bottom. Then divide.
$10^2 = 10 \times 10 = 100$
$\frac{1}{100} = 0.01$
One hundredth is 0.01, with the 1 two places after the decimal point.
Watch out: 0.1 is one tenth, not one hundredth. That slip gives choice C. Adding 100 instead of $\frac{1}{100}$ gives choice D.
Add the four terms
Line up the decimal points and add. The 0.01 goes in the hundredths place.
$5{,}000 + 30 + 9 + 0.01 = 5{,}039.01$
So the answer is B, 5,039.01.

Check your answer

The first three terms are whole numbers. They add up to 5,039. The last term is a small part of 1. So the answer is just over 5,039. Only B and C fit. The last term is hundredths, so B is right.

The original DOE question

Question 91 as printed on page 159 of the DOE handbook. — As printed in the DOE’s Specialized High Schools Student Handbook 2018–2019, page 159, question 91. Open page 159 of the handbook (PDF)