SHSAT Sunset

(5⁴ × 5⁶)/5²

2019 Handbook

2019 Test B Math · Question 22 of 57

Question

54×5652=\frac{5^{4} \times 5^{6}}{5^{2}} =

  • A585^{8}
  • B5105^{10}
  • C5125^{12}
  • D5225^{22}
Show the answer

Correct answer: A

54×5652=51052=58\frac{5^{4} \times 5^{6}}{5^{2}} = \frac{5^{10}}{5^{2}} = 5^{8}

Question analysis

This exponents question asks for the value of 5 to the 4th times 5 to the 6th, divided by 5 squared, written as a single power of 5. It tests the rules for multiplying and dividing powers with the same base. Every choice is a power of 5, so the whole question rests on combining the exponents correctly.

NYS standards
  • 8.EE.1 integer exponent properties
Grade level
Grade 8 content, by NYS standard
Compared with grade 7
Above grade 7 Combining powers of one base by exponent rules is grade 8 content; grades 6 and 7 only evaluate powers. The task itself is short, one product and one quotient of powers of 5.
Estimated difficulty
Straightforward: two steps or one careful reading.
What makes it hard
Solution
2 steps
Reading level
Question and choices: about grade 16+, 5 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

Everything rests on two grade 8 rules, and the choices are built so that each misapplied rule has an answer. Every choice is a power of 5, so the base gives nothing away and only the exponent arithmetic decides. Students who carry the written operations onto the exponents, or who stop after the numerator, find their result listed; working the top first, then the bottom, keeps the two rules apart.

  1. Something you must already know moderate Working it out

    Neither exponent rule is given: multiplying 5⁴ by 5⁶ adds the exponents to 10, and dividing by 5² subtracts 2; working out the powers instead, 625 × 15,625, is hopeless by hand. (Anderson & Krathwohl, 2001)

Strategy: When powers of the same base are multiplied, add their exponents; when one is divided by another, subtract the exponent of the divisor. Apply the top first, then the bottom.

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.
  • Stops one step early: A choice is a number from partway through the solution.

The wrong answers, classified

How the questions are analyzed, and the research behind it

Worked solution, step by step

Formulas used

Multiplying powers with the same base
am×an=am+na^{m} \times a^{n} = a^{m + n}
a is the base, the number being multiplied. m and n are exponents, the small raised numbers. Each exponent counts how many a’s are multiplied. Putting the two groups together makes m + n of them, so you add the exponents.
Dividing powers with the same base
aman=am−n\frac{a^{m}}{a^{n}} = a^{m - n}
a is the base, and 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, and you subtract the exponents.

  1. Sort out the question

    The expression has three powers of 5. A power like 545^{4} means 5 multiplied by itself 4 times: 5×5×5×55 \times 5 \times 5 \times 5. The 5 is the base. The small raised 4 is the exponent. Every choice is a power of 5, so you only need the final exponent.

  2. Multiply the powers on top

    The top multiplies two powers of 5. Use the rule for multiplying powers.

    am×an=am+na^{m} \times a^{n} = a^{m + n}

    54×56=54+6=5105^{4} \times 5^{6} = 5^{4 + 6} = 5^{10}

    Four 5s times six more 5s makes ten 5s in a row.

    Watch out: add the exponents. Don’t multiply them. 4×6=244 \times 6 = 24 leads to 5225^{22}, choice D.

  3. Divide by the power on the bottom

    Now divide by the bottom. Use the rule for dividing powers.

    aman=am−n\frac{a^{m}}{a^{n}} = a^{m - n}

    51052=510−2=58\frac{5^{10}}{5^{2}} = 5^{10 - 2} = 5^{8}

    Two of the ten 5s on top cancel with the two 5s on the bottom. Eight 5s are left.

    Watch out: stopping at 5105^{10}, choice B, forgets the bottom. Multiplying and dividing the exponents, 4×6÷2=124 \times 6 \div 2 = 12, gives choice C.

    So the answer is A, 585^{8}.

Check your answer

Count the 5s. The top has 4+6=104 + 6 = 10 fives multiplied together. The bottom has 2. Each 5 on the bottom cancels one 5 on top, since 5 ÷ 5 = 1. That leaves 10−2=810 - 2 = 8 fives, which is 5 to the 8th power.

More questions like this

The original DOE question

Question 79 as printed on page 180 of the DOE handbook.
As printed in the DOE’s Specialized High Schools Student Handbook 2019–2020, page 180, question 79. Open page 180 of the handbook (PDF)