0³ + 4⁰ + 2⁻¹ + (−1)²

2023 Handbook

2023 Test B Math · Question 46 of 57

Question

$0^{3}+4^{0}+2^{-1}+(-1)^{2}$

What is the value of the expression above, in decimal form?

A0
B0.5
C1.5
D2.5

Show the answer

Correct answer: D

$0^{3}+4^{0}+2^{-1}+(-1)^{2}$

$=0+1+\frac{1}{2}+1$

$=2\frac{1}{2}=2.5$

Question analysis

An exponents question: students find the decimal value of a sum of four powers, zero cubed, four to the zero power, two to the negative first power and negative one squared. Each of the four terms follows a different rule, and a slip on any one of them changes the total.

NYS standards: 8.EE.1 integer exponent properties
7.NS.2a rational number multiplication
Grade level: Grade 8 content, by NYS standard
Compared with grade 7: Above grade 7 Grade 7 work uses whole-number exponents and signed numbers. Zero and negative exponents are grade 8 content, and this expression tests both alongside a negative base, each with its own common misconception.
Estimated difficulty: Hard: combines ideas, needs a plan, and has more than one tempting wrong answer.
Thinking it demands: Depth of knowledge 1, recall and reproduction · mainly apply
What makes it hard: Something you must already know major
Error-prone arithmetic minor
Solution: 4 steps
Reading level: Question and choices: about grade 6, 12 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

Every term tests a different exponent rule, and each wrong choice comes from breaking exactly one of them. The zero exponent is the most common casualty, since 4⁰ looks as if it should be 0 like 0³; a negative exponent read as a negative number is close behind. Writing each term’s value above it before adding, 0, 1, 1/2 and 1, keeps the four rules separate.

Something you must already know major Working it out
Three rules the test does not give decide the sum: any nonzero number to the 0 power is 1, 2 to the −1 power is 1/2, and (−1)² is positive 1. (Anderson & Krathwohl, 2001)
Error-prone arithmetic minor Working it out
The four terms carry a zero base, a zero exponent, a negative exponent and a negative base, so a sign or a stray 0 can slip in at each one. (Brown & Burton, 1978; Junker & Sijtsma, 2001)

Strategy: Evaluate each power on its own, using the rules for a zero exponent and a negative exponent and the sign of a squared negative number, then add the four values.

Traps

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

The wrong answers, classified

A A faulty procedure (Brown & Burton, 1978; Tatsuoka, 1983)
Reveals a misconception Expected pull: moderate
Treats 2 to the power −1 as −2 instead of one half, so the four terms 0 + 1 − 2 + 1 add to 0.
B Order of operations (Brown & Burton, 1978)
Reveals a misconception Expected pull: weak
Squares as if the minus sat outside the parentheses, taking (−1)² as −1, so the terms add to 0 + 1 + 0.5 − 1 = 0.5.
C A faulty procedure (Brown & Burton, 1978; Tatsuoka, 1983)
Reveals a misconception Expected pull: strong
Takes 4 to the power 0 as 0 rather than 1, so the expression becomes 0 + 0 + 0.5 + 1 = 1.5.

How the questions are analyzed, and the research behind it

Worked solution, step by step

Formulas used

Zero exponent: $a^0 = 1$; a is any number except 0. An exponent of 0 always gives 1. It works because each step down in the exponent divides by a. For example, 4 to the first power is 4, and 4 ÷ 4 = 1.
Negative exponent: $a^{-n} = \frac{1}{a^n}$; a is the base, and n is a positive whole number. A negative exponent does not make the number negative. It means 1 divided by the base to the positive power. For example, 2 to the −1 power is one half.

Sort out the question
The expression adds four powers. A power is a base with an exponent, a small raised number. The exponent tells how many times to use the base as a factor, a number you multiply. Find each power, then add.
Find the first two powers
Zero times itself is still zero.
$0^3 = 0 \times 0 \times 0 = 0$
Next, use the zero exponent rule.
$4^0 = 1$
Watch out: $4^0$ is 1, not 0. Taking it as 0 gives choice C.
Find the negative power
Use the negative exponent rule.
$2^{-1} = \frac{1}{2^1} = \frac{1}{2} = 0.5$
Watch out: $2^{-1}$ is not −2. That mistake gives choice A.
Find the last power
The base is −1, inside the parentheses. Multiply it by itself. A negative times a negative is positive.
$(-1)^2 = (-1) \times (-1) = 1$
Watch out: The answer is not −1. That mistake gives choice B.
Add the four values
Now add them. The question wants a decimal, so write one half as 0.5.
$0 + 1 + 0.5 + 1 = 2.5$
So the answer is D, 2.5.

Check your answer

See why $4^0 = 1$ with a pattern. $4^2 = 16$ and $4^1 = 4$ . Each step down divides by 4, so $4^0 = 4 \div 4 = 1$ . In the same way, $2^0 = 1$ and $2^{-1} = 1 \div 2 = \frac{1}{2}$ .

The original DOE question

Question 103 as printed on page 204 of the DOE handbook. — As printed in the DOE’s NYC Guide to the Specialized High Schools Admissions Test (SHSAT) for 2024 Admissions, page 204, question 103. Open page 204 of the handbook (PDF)