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8th Grade Math — 12-Lesson Comprehensive Quiz

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1.Which option correctly writes 10×10×10×10×10×10 in both exponential form and standard form?

10⁶ = 100,00010⁵ = 1,000,00010⁷ = 1,000,00010⁶ = 1,000,000

2.A data center needs to express 10,000,000 as a power of 10. Which form should it use?

10⁷10⁶10⁸7¹⁰

3.Select all the correct statements about 10⁴.

Its expanded form contains 4 factors of 10 multiplied togetherIts standard form is 10,000In standard form, the 1 is followed by 4 zerosIt represents 10+10+10+10

4.A student says, 'The standard form of 10⁹ is 900,000,000 because the exponent is 9.' Identify the error and write the correct standard form.

5.Calculate 6.38×10³. Which option gives both the correct reasoning and the correct result?

Move the decimal point 2 places to the right to get 638Simply add 3 zeros after 6.38 to get 6.38000Move the decimal point 3 places to the left to get 0.00638Move the decimal point 3 places to the right to get 6,380

6.What is the result of 72.5÷10²?

7,2507.250.07250.725

7.Fill in the blank: 0.0049×10⁴ = ____.

8.A student calculates 0.84÷10³ and gets 840. Diagnose the error by deciding whether the result should increase or decrease, then give the correct answer.

9.Given 3.27×10ⁿ=32,700, find n and explain how you determined it.

10.Given 3⁻ⁿ=1/729, where n is a positive integer, find n and explain your reasoning.

11.Calculate: what is $10^{-2}$?

1/100−1001/20100

12.Calculate: $4^{-3}=\underline{\hspace{3cm}}$. Give your answer as a fraction in simplest form.

13.A student says $2^{-5}=-32$: because the exponent is negative, the answer must be negative. Which evaluation is most accurate?

Correct: since 2^5=32, simply add a minus signIncorrect: a negative exponent always gives 0Correct: a negative exponent makes the result negativeIncorrect: a negative exponent means taking the reciprocal, so $2^{-5}=1/32$, which is not negative

14.Which equation is correct?

$5^{-3}=1/125$$5^{-3}=125$$5^{-3}=1/15$$5^{-3}=−125$

15.Select all the correct descriptions of how to calculate $12^{-2}$.

It can be rewritten as $1/12^2$The base 12 stays unchangedThe result is $1/144$The result is −144After rewriting, the exponent −2 becomes positive 2

16.Compare $7^2$ and $7^{-2}$. Which statement is correct?

$7^2=1/49$, while $7^{-2}=49$$7^2=49$, while $7^{-2}=1/49$; the negative exponent gives a smaller result, but not a negative oneBoth equal 49 because the sign of the exponent does not affect the result$7^{-2}=-49$ because the exponent is negative

17.Using only the number 7, the exponent 2, and one minus sign, write a power expression equal to 49 and another equal to −49. Explain the role of parentheses.

18.A student notices that (-4)³ and -4³ both equal −64, then claims, 'Parentheses never affect the result.' Evaluate the claim and give a counterexample using an exponent of 2.

19.According to the rules in the video, which statements are correct? Select all that apply.

$(-2)^3=-8$$(-3)^4=-81$$-2^3=-8$$(-10)^3=1000$

20.Using the video's method of multiplying the base by itself, what is $(-4)^3$?

$-64$$64$$-12$$12$

21.In one sentence, explain why $(-3)^4$ is positive, and give its value.

22.Which option most accurately describes a terminating decimal?

It must be an integerIt cannot contain a decimal pointIts decimal digits end after a finite number of placesIts decimal digits repeat a pattern forever

23.What is the key feature of a repeating decimal?

A digit or group of digits repeats in a pattern foreverIt can have only one decimal placeDivision immediately gives a remainder of 0It must be greater than 1

24.When converting a fraction to a decimal, what is the correct order of division?

Numerator × denominatorNumerator + denominatorDenominator ÷ numeratorNumerator ÷ denominator

25.When calculating 20 ÷ 9, the remainder returns to 2 after each subtraction. What can you conclude about the result?

You cannot tell because no negative number has appearedIt is a repeating decimal because the same division steps keep repeatingIt is an integer because its integer part is 2It is a terminating decimal because the remainder is already smaller than the divisor

26.Which option gives the decimal form and type of 20/9?

2.2, a terminating decimal0.222…, a repeating decimal2.222…, a repeating decimal, with a bar over the decimal digit 22.25, a terminating decimal

27.Which option correctly converts 3/8 to a decimal?

0.375375…, a repeating decimal0.0375, a terminating decimal0.375, with the digit 5 repeating forever0.375, a terminating decimal

28.A student says, 'The first step of long division did not give a remainder of 0, so the result must be a repeating decimal.' Explain why this is incorrect, using the calculation of 3/8 as evidence.

29.Where should the bar be placed in bar notation?

Only above the first digit after the decimal pointAbove the digit or group of digits that repeats foreverAbove the entire number, including the integer partOnly above the last digit written

30.Which option correctly writes 0.4444… in bar notation?

Place a bar above both 0 and 4Place a bar only above 0Write 0.4 with a bar only above 4Write 0.4 without a bar

31.What is the repeating block in 0.363636…?

3663363

32.When writing 0.83333… in bar notation, which digits should the bar cover?

0, 8, and 3Only 3Only 8Both 8 and 3

33.A student writes 4.272727… as 4.27 with a bar only above 7. Identify the error and write the correct bar notation.

34.Calculate (0.3)⁴. Which option gives both the correct expanded form and the correct result?

0.3+0.3+0.3+0.3=1.20.3×0.3×0.3×0.3=0.00810.3×4=1.20.3×0.3×0.3×0.3=0.81

35.Calculate (0.06)³, write its expanded form, and explain how to determine the position of the decimal point in the result.

36.A student calculates (1.4)² by first finding 14×14=196, then writing 19.6. Identify the decimal-point error and give the correct result and expanded form.

37.A student calculates (0.2)⁴ as 0.8, saying that the exponent 4 means multiplying 0.2 by 4. Which correction is most accurate?

Multiply four factors of 0.2; 2⁴=16, and there are 4 decimal places in total, giving 0.0016Calculate 2⁴=16, then keep 1 decimal place to get 1.6Add 4 to 0.2 to get 4.2Calculate 2⁴=16 and ignore the decimal point

38.Calculate (3/5)³. Which option gives both the correct expanded form and the correct result?

(3/5)×(3/5)×(3/5)=27/125(3×3×3)/(5+5+5)=27/15(3/5)×3=9/5(3×3×3)/5=27/5

39.Calculate (1/3)⁴, write its expanded form, and explain why the numerator of the result is still 1.

40.Calculate (6/8)². First give the unsimplified fraction, then reduce it to simplest form and explain what you did to the numerator and denominator.

41.A student writes (2/10)⁴ as 16/40. What is the error, and which result is correct?

There is no error; 16/40=2/5Only the denominator should be raised to the power; the correct result is 2/10000The numerator should be 2×4; the correct result is 8/40The fourth power of the denominator was confused with multiplying it by 4; the correct result is 16/10000=1/625

42.Use long division to calculate 17÷9. Give the integer part, then the quotient digit and remainder for at least two decimal steps. Explain why the decimal repeats and express the result in bar notation.

43.Convert 5/11 to a decimal. Show the division steps that produce the first four decimal digits, including the quotient digit and remainder at each step. Identify the repeating block and explain which digits the bar should cover.

44.While calculating 14÷9, a student has written 1.555 and notices that the remainder always returns to 5. The student says, 'I have calculated three decimal places, so the exact answer is 1.555.' Which evaluation is correct?

Incorrect: the same remainder makes the steps keep repeating; the exact answer is 1.555…, with a bar over the decimal digit 5Correct: calculating three decimal places means the decimal endsCorrect: because remainder 5 is less than divisor 9, division should stopIncorrect: the exact answer is 1.556 because all repeating decimals must be rounded up

Variant question

This is an adapted exercise. The linked video reviews the underlying concept and may use different numbers, examples, or wording; it may not contain this exact question.

45.A student thinks 8⁻² and (-8)² are equal because both contain a minus sign and an exponent of 2. Evaluate both expressions, explain the different roles of the minus signs, and identify which result is greater than 1.

Variant question

This is an adapted exercise. The linked video reviews the underlying concept and may use different numbers, examples, or wording; it may not contain this exact question.

46.A student says, '(-5)³ and -5³ both equal −125, so (-5)⁴ and -5⁴ must also be equal.' Evaluate the two fourth-power expressions, explain the error using the order of operations, and explain why the cubes happen to be equal.

Variant question

This is an adapted exercise. The linked video reviews the underlying concept and may use different numbers, examples, or wording; it may not contain this exact question.

47.Digits in parentheses indicate the repeating block covered by the bar. Compare 9.(752) and 9.7(52): write the first 9 digits after the decimal point for each, decide whether they represent the same number, and explain why the 7 in the second number should not be covered by the bar.

48.Simplify (x³)². Use expansion into identical factors to explain why the exponents are multiplied rather than added.

49.A student writes (2²)⁴ as 2⁶. Identify the rule error, give the correct exponential and standard forms, and verify them by expansion.

50.A student writes x⁴×x³=x¹². Correct the error and explain the correct rule by counting the factors in the expanded form.

51.Write 5⁴×5² as a single power and evaluate it. Which option is correct?

5⁶=1562525⁶=2441406255⁸=39062510⁶=1000000

Introduction to Exponents: Meaning, Expansion, and Calculation

52.A student says, 'In 6², the base is 2 and the exponent is 6, so 6²=6×2=12.' Correct the identification of the base and exponent, write the correct expanded form and result, and explain what the exponent 2 means.

53.How do you expand and evaluate '4 squared' and '4 cubed', respectively?

4×2=8; 4×3=124×4×4=64; 4×4=164+4=8; 4+4+4=124×4=16; 4×4×4=64

54.Write 2×2×2×2×2×2 in exponential form and identify the base and exponent. Then evaluate it by pairing the six factors. Explain why this gives three factors of 4 multiplied together, not three 4s added together.

55.Calculate 3⁵ and 6⁴, writing the expanded form of each. Show intermediate results using successive multiplication for 3⁵ and paired factors for 6⁴. A student stops at 81 when calculating 3⁵; identify what was omitted.