Friday, May 7, 2021

Grade 8 PARCC math: integer exponents


The following multiple-select question, explained here in hopes of helping eighth-grade students and their parents in Maryland and Illinois prepare for the PARCC test near the end of this school year, appears on the released version of PARCC’s Spring 2016 test for grade eight math:

Which expression is equivalent to (7^3)^5 \cdotp 7^4 ?

Select each correct answer.

  • A.   7^{3 \cdotp 5 \cdotp 4}
  • B.   7^{3 \cdotp 5 + 4}
  • C.   7^{3+5+4}
  • D.   7^{3(5+4)}
  • E.   7^{3 \cdotp 5} \textrm{ }\cdotp 7^4
  • F.   7^{3 + 5} \textrm{ }\cdotp 7^4

Answer and references

Correct answer: B and E.

Common Core Math Content 8th grade, Expressions and Equations, Work with radicals and integer exponents:

(8.EE.A.1) Know and apply the properties of integer exponents to generate equivalent numerical expressions. For example,

3^2 \times 3^{-5} = 3^{-3} = (\frac{1}{3})^{3} = \frac{1}{27}

Solution strategy (there are others)

Apply the rules for exponents to each of the answer choices, since the base is the same in all cases: 7.

Any number a raised to a power mn equals

a^{m \cdotp n}

In other words,

(7^3)^5 = 7^{3 \cdotp 5}

And any number a raised to a power m + n equals

a^m \cdotp a^n

In other words,

a^{m+n} = a^m \cdotp a^n \textrm{ or } 7^{15} \cdotp 7^4 = 7^{15+4}

That means that any expression that equals 719 will be a match.

Consider A:   7^{3 \cdotp 5 \cdotp 4} = 7^{60} \textrm{ which is incorrect.}

Consider B:   7^{3 \cdotp 5 + 4} = 7^{19} \textrm{ BINGO.}

Consider C:   7^{3+5+4} = 7^{12} \textrm{ nope.}

Consider D:   7^{3(5+4)} = 7^{3 \cdotp 9} = 7^{27} \textrm{ not happening.}

Consider E:   7^{3 \cdotp 5} \textrm{ }\cdotp 7^4 = 7^{15} \cdotp 7^4 = 7^{15+4} \textrm{ works.}

Consider F:   7^{3 + 5} \textrm{ }\cdotp 7^4 = 7^8 \cdotp 7^4 = 7^{12} \textrm{ not even close.}

Analysis of this question and online accessibility

Integer exponents with numerical bases are new for eighth graders under the Common Core. Students should know how to work with positive and negative exponents of a numerical base (fraction or integer), as well as with exponents that are themselves expressions.

If you or your parents need some practice, check out this worksheet with answers for the power rule and this one with more complete coverage of all the rules for exponents.

Multiple-choice questions like this can be delivered easily online and on paper-based tests, which makes this question accessible for students on any device they may use or on paper. Validity, reliability, and fairness measures should not differ significantly among the various delivery modes.

No special accommodation challenges can be identified with this question, so the question is considered fair.

Paul Katula
Paul Katula is the executive editor of the Voxitatis Research Foundation, which publishes this blog. For more information, see the About page.

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