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AK STAR Math Grade 9 study materials
AK STAR Math Grade 9 study resources

AK STAR Math Grade 9 practice questions

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Question 1Number and Quantity

Which of the following numbers is an irrational number?

  1. −3/4
  2. √7 Correct answer.
  3. √81
  4. 0.75

Explanation

A rational number can be written as a ratio of two integers, and its decimal form either terminates or repeats. √81 = 9, 0.75, and −3/4 are all rational. Since 7 is not a perfect square, √7 has a decimal expansion that never terminates and never repeats.

Question 2Number and Quantity

Which expression is equal to 3^4 · 3^5?

  1. 9^9
  2. 3^1
  3. 3^9 Correct answer.
  4. 3^20

Explanation

When multiplying powers that share the same base, keep the base and add the exponents: 3^4 · 3^5 = 3^(4+5) = 3^9.

Question 3Number and Quantity

Which number is 4.2 × 10^3 written in standard form?

  1. 420
  2. 42
  3. 4200 Correct answer.
  4. 42000

Explanation

A positive exponent of 3 moves the decimal point three places to the right. Starting with 4.2, that gives 42, then 420, then 4200.

Question 4Number and Quantity

What is the value of √(144/25)?

  1. 72/25
  2. 12/5 Correct answer.
  3. 144/5
  4. 12/25

Explanation

The square root of a quotient of perfect squares equals the square root of the numerator divided by the square root of the denominator: √144 / √25 = 12/5.

Question 5Number and Quantity

Which property of real numbers is illustrated by 8 + (−8) = 0?

  1. identity property of addition
  2. commutative property of addition
  3. associative property of addition
  4. inverse property of addition Correct answer.

Explanation

The inverse property of addition states that every number a has an opposite −a such that a + (−a) = 0. Here 8 and −8 are opposites whose sum is zero.

Question 6Number and Quantity

What is ∛(−125)?

  1. not a real number
  2. 5
  3. −5 Correct answer.
  4. −25

Explanation

A cube root asks for a number that produces the radicand when used as a factor three times. Because (−5)(−5)(−5) = −125, the cube root is −5, and odd roots of negative numbers are real.

Question 7Number and Quantity

How is 0.00037 written in scientific notation?

  1. 3.7 × 10^5
  2. 3.7 × 10^−4 Correct answer.
  3. 3.7 × 10^3
  4. 3.7 × 10^4

Explanation

Move the decimal point right until exactly one nonzero digit sits in front of it: 0.00037 becomes 3.7, which requires 4 places. Moving right makes the exponent negative, giving 3.7 × 10^−4.

Question 8Number and Quantity

Between which two consecutive integers does √40 lie?

  1. 5 and 6
  2. 7 and 8
  3. 6 and 7 Correct answer.
  4. 4 and 5

Explanation

Compare 40 with nearby perfect squares: 6^2 = 36 and 7^2 = 49. Since 36 < 40 < 49, it follows that 6 < √40 < 7.

Question 9Number and Quantity

Which expression has the same value as 1/64?

  1. 2^−5
  2. 2^−6 Correct answer.
  3. −2^6
  4. 2^6

Explanation

A negative exponent means the reciprocal of the matching positive power, so 2^−6 = 1/2^6 = 1/64.

Question 10Number and Quantity

Which expression has a value that is a rational number?

  1. √3 + 1
  2. √5 · √10
  3. √2 · √8 Correct answer.
  4. π + 2

Explanation

√2 · √8 = √16 = 4, and 4 is an integer, so the value is rational. The other expressions still contain an irrational part after simplifying.

Question 11Number and Quantity

A car travels at a constant speed of 60 miles per hour. What is this speed in feet per second? (1 mile = 5,280 feet)

  1. 44 ft/s
  2. 132 ft/s
  3. 352 ft/s
  4. 88 ft/s Correct answer.

Explanation

Convert both units in one chain: (60 mi/h)(5,280 ft/1 mi)(1 h/3,600 s) = 316,800 ÷ 3,600 = 88 ft/s.

Question 12Number and Quantity

Which of these numbers is rational?

  1. 22/7 Correct answer.
  2. ∛9
  3. π/2
  4. √11

Explanation

22/7 is a ratio of two integers, so it is rational, even though it is often used as a handy approximation of π. The other three cannot be written as ratios of integers.

Question 13Number and Quantity

Simplify (3x^2)^3.

  1. 27x^6 Correct answer.
  2. 9x^6
  3. 9x^5
  4. 27x^5

Explanation

Raise every factor inside the parentheses to the third power: 3^3 = 27 and (x^2)^3 = x^(2·3) = x^6. The simplified result is 27x^6.

Question 14Number and Quantity

Which list shows these numbers in order from least to greatest?

  1. 2.9, √8, 10/3, π
  2. 2.9, √8, π, 10/3
  3. √8, 2.9, π, 10/3 Correct answer.
  4. √8, 2.9, 10/3, π

Explanation

Using decimal approximations: √8 ≈ 2.83, 2.9 stays 2.9, π ≈ 3.14, and 10/3 ≈ 3.33. Comparing the decimals gives √8 < 2.9 < π < 10/3.

Question 15Number and Quantity

What is the value of (1/3)^−2 + 2^0?

  1. 8
  2. −7
  3. 10 Correct answer.
  4. 6

Explanation

A negative exponent flips the fraction, so (1/3)^−2 = 3^2 = 9. Any nonzero number raised to the zero power equals 1, so the sum is 9 + 1 = 10.

Question 16Number and Quantity

How many times as large as 4 × 10^2 is 8 × 10^6?

  1. 4 × 10^4
  2. 2 × 10^8
  3. 2 × 10^3
  4. 2 × 10^4 Correct answer.

Explanation

Divide the coefficients and subtract the exponents of the powers of ten: (8 ÷ 4) × 10^(6−2) = 2 × 10^4.

Question 17Number and Quantity

Which integer is closest to ∛50?

  1. 3
  2. 4 Correct answer.
  3. 6
  4. 5

Explanation

Because 3^3 = 27 and 4^3 = 64, the cube root of 50 lies between 3 and 4. The value 50 is much closer to 64 than to 27, so the closest integer is 4.

Question 18Number and Quantity

Which expression equals 24 + 36 and is written using the greatest common factor?

  1. 6(4 + 6)
  2. 4(6 + 9)
  3. 12(2 + 3) Correct answer.
  4. 2(12 + 18)

Explanation

The greatest common factor of 24 and 36 is 12. Writing each addend as a multiple of 12 gives 24 + 36 = 12 · 2 + 12 · 3 = 12(2 + 3).

Question 19Number and Quantity

Simplify (5a^2b^0)(−2a^3b), assuming a and b are nonzero.

  1. 10a^5b
  2. −10a^6b
  3. −10a^5b^0
  4. −10a^5b Correct answer.

Explanation

Since b^0 = 1, the expression is (5a^2)(−2a^3b). Multiply the coefficients to get −10, add the exponents on a to get a^5, and keep b to the first power.

Question 20Number and Quantity

A student claims, "The sum of two irrational numbers is always irrational." Which expression is a counterexample?

  1. π + (−π) Correct answer.
  2. √3 + √2
  3. √9 + √4
  4. √2 + √2

Explanation

Both π and −π are irrational, but their sum is 0, which is rational. One counterexample is enough to show that the student's claim is false.

Question 21Number and Quantity

Simplify (2^3 · 2^−5)^−2.

  1. −1/16
  2. 1/16
  3. −16
  4. 16 Correct answer.

Explanation

Inside the parentheses, add the exponents: 2^(3 + (−5)) = 2^−2. Then apply the outer power by multiplying exponents: (2^−2)^−2 = 2^4 = 16.

Question 22Number and Quantity

A single grain of sand has a mass of about 5 × 10^−5 kilogram. A small bucket holds about 4 × 10^6 grains of sand. What is the approximate total mass of the sand in the bucket, in kilograms?

  1. 2,000 kg
  2. 20 kg
  3. 200 kg Correct answer.
  4. 2 kg

Explanation

Multiply the mass of one grain by the number of grains: (5 × 10^−5)(4 × 10^6) = 20 × 10^1 = 2 × 10^2, which is 200 kilograms.

Question 23Number and Quantity

Which number is greatest?

  1. ∛30
  2. 3.1
  3. √10 Correct answer.
  4. 22/7

Explanation

Approximating each value gives ∛30 ≈ 3.107, 22/7 ≈ 3.143, 3.1, and √10 ≈ 3.162. The largest of these approximations is 3.162, so √10 is the greatest number.

Question 24Number and Quantity

A faucet drips 3 drops per second, and each drop has a volume of 0.05 milliliter. How many liters does the faucet waste in one day? (1 liter = 1,000 milliliters; 1 day = 86,400 seconds)

  1. 1296 L
  2. 12.96 L Correct answer.
  3. 0.1296 L
  4. 1.296 L

Explanation

In one day the faucet drips 3 × 86,400 = 259,200 drops, and the total volume is 259,200 × 0.05 = 12,960 milliliters. Dividing by 1,000 converts to 12.96 liters.

Question 25Number and Quantity

Which statement about the value of √50 + √18 is true?

  1. It lies between 8 and 9.
  2. It equals 8.
  3. It lies between 12 and 13.
  4. It lies between 11 and 12. Correct answer.

Explanation

√50 ≈ 7.07 and √18 ≈ 4.24, so their sum is about 11.31. Equivalently, 5√2 + 3√2 = 8√2 ≈ 11.31, which falls between 11 and 12.

Question 26Number and Quantity

Which expression has the same value as 2^3 · 2^2?

  1. 2^6
  2. 2^5 Correct answer.
  3. 4^5
  4. 2^1

Explanation

When multiplying powers with the same base, add the exponents: 2^3 · 2^2 = 2^(3+2) = 2^5. The base stays 2.

Question 27Number and Quantity

Which number is written correctly in scientific notation?

  1. 0.56 × 10^4
  2. 56 × 10^2
  3. 5.6 × 10^3 Correct answer.
  4. 560 × 10^1

Explanation

In scientific notation, the first factor must be at least 1 and less than 10, and the second factor must be a power of 10. Only 5.6 × 10^3 fits that form.

Question 28Number and Quantity

Which fraction has a decimal expansion that terminates?

  1. 5/6
  2. 2/3
  3. 7/9
  4. 3/8 Correct answer.

Explanation

A fraction in lowest terms has a terminating decimal exactly when its denominator has no prime factors other than 2 and 5. Since 3/8 has denominator 8 = 2^3, its decimal terminates (0.375).

Question 29Number and Quantity

What is the value of √(49/64)?

  1. 7/8 Correct answer.
  2. 7/64
  3. 7/32
  4. 49/8

Explanation

The square root of a quotient equals the quotient of the square roots when both are nonnegative: √(49/64) = √49 / √64. That gives 7/8.

Question 30Number and Quantity

What is the value of 16^(3/4)?

  1. 64
  2. 12
  3. 4
  4. 8 Correct answer.

Explanation

For a^(m/n), take the nth root first and then raise to the mth power. Since 16^(1/4) = 2, we get 16^(3/4) = 2^3 = 8.

Question 31Number and Quantity

Which expression is equivalent to √72?

  1. 36√2
  2. 3√6
  3. 6√2 Correct answer.
  4. 2√6

Explanation

Factor 72 to reveal a perfect square: 72 = 36·2. Then √72 = √36·√2 = 6√2.

Question 32Number and Quantity

Which expression is equivalent to 8/√2?

  1. √2/4
  2. 8√2
  3. 4√2 Correct answer.
  4. 4

Explanation

Multiply numerator and denominator by √2 to remove the radical from the denominator: (8/√2)(√2/√2) = 8√2/2 = 4√2.

Question 33Number and Quantity

What is (8.4 × 10^9) ÷ (2.1 × 10^3) in scientific notation?

  1. 6.3 × 10^6
  2. 4 × 10^3
  3. 4 × 10^6 Correct answer.
  4. 4 × 10^12

Explanation

Divide the coefficients and subtract the exponent in the denominator from the exponent in the numerator: 8.4/2.1 = 4 and 10^9/10^3 = 10^(9−3) = 10^6. The result is 4 × 10^6.

Question 34Number and Quantity

What is (3.5 × 10^5) + (2.4 × 10^6) written in scientific notation?

  1. 5.9 × 10^6
  2. 2.75 × 10^6 Correct answer.
  3. 5.9 × 10^11
  4. 2.75 × 10^5

Explanation

Rewrite 3.5 × 10^5 as 0.35 × 10^6 so both terms have the same power of ten. Then 0.35 × 10^6 + 2.4 × 10^6 = 2.75 × 10^6.

Question 35Number and Quantity

Which fraction is equal to the repeating decimal 0.272727...?

  1. 3/11 Correct answer.
  2. 11/3
  3. 27/100
  4. 27/1000

Explanation

Let x = 0.272727... Then 100x = 27.272727..., so 100x − x = 27. This gives 99x = 27, or x = 27/99 = 3/11.

Question 36Number and Quantity

For x > 0, which expression is equivalent to x^(2/3)?

  1. x^2/3
  2. √(x^3)
  3. x^(3/2)
  4. ∛(x^2) Correct answer.

Explanation

For x^(m/n), the denominator n is the index of the radical and the numerator m is the exponent inside it. Thus x^(2/3) = ∛(x^2).

Question 37Number and Quantity

Simplify (√5 + √2)(√5 − √2).

  1. 3 Correct answer.
  2. 3 + 2√10
  3. 7
  4. 10

Explanation

This product matches (a+b)(a−b) = a^2 − b^2 with a = √5 and b = √2. So the value is (√5)^2 − (√2)^2 = 5 − 2 = 3.

Question 38Number and Quantity

If x > 0, which expression is equivalent to (x^(1/2) · x^(1/3))^6?

  1. x^6
  2. x^36
  3. x^5 Correct answer.
  4. x^1

Explanation

First add exponents inside the parentheses: x^(1/2)·x^(1/3) = x^(1/2+1/3) = x^(5/6). Raising to the sixth power gives x^(5/6·6) = x^5.

Question 39Number and Quantity

A rectangle has area 3 × 10^4 square centimeters and length 2 × 10^2 centimeters. What is its width in meters?

  1. 150 m
  2. 1.5 m Correct answer.
  3. 0.15 m
  4. 15 m

Explanation

Divide area by length: (3 × 10^4)/(2 × 10^2) = 1.5 × 10^2 centimeters. Since 100 centimeters = 1 meter, 1.5 × 10^2 cm = 150 cm = 1.5 m.

Question 40Number and Quantity

A factory machine produces 6.0 × 10^4 parts per hour. At this constant rate, how many parts does it produce per second?

  1. 25/3 parts
  2. 50/3 parts Correct answer.
  3. 5/3 parts
  4. 500/3 parts

Explanation

Convert the hourly rate to a per-second rate by dividing by the number of seconds in one hour: 6.0 × 10^4 ÷ 3,600 = 60,000/3,600 = 50/3. This is about 16.7 parts per second.

Question 41Algebra

Solve for x: 3x − 7 = 14

  1. x = 7/3
  2. x = 21
  3. x = 3
  4. x = 7 Correct answer.

Explanation

Undo the subtraction first by adding 7 to both sides, which gives 3x = 21. Then divide both sides by the coefficient 3 to isolate x, so x = 7. Checking back: 3(7) − 7 = 14, which is true.

Question 42Algebra

Which expression is equivalent to x⁵ · x³?

  1. x¹⁵
  2. x⁸ Correct answer.
  3. x²
  4. 2x⁸

Explanation

When multiplying powers that have the same base, keep the base and add the exponents: x⁵ · x³ = x^(5+3) = x⁸. Multiplying the exponents or adding the coefficients are different rules that do not apply here.

Question 43Algebra

What is the slope of the line that passes through the points (2, 3) and (6, 11)?

  1. 2 Correct answer.
  2. 4
  3. 1/2
  4. 8

Explanation

Slope equals the change in y divided by the change in x: (11 − 3)/(6 − 2) = 8/4 = 2. The order of subtraction must be consistent in both the numerator and the denominator.

Question 44Algebra

If f(x) = 2x² − 1, what is f(3)?

  1. 17 Correct answer.
  2. 5
  3. 18
  4. 35

Explanation

Substitute 3 for every x: f(3) = 2(3)² − 1. Square first to get 9, then multiply by 2 to get 18, and subtract 1, giving 17.

Question 45Algebra

Solve the inequality −2x > 10.

  1. x > 5
  2. x > −5
  3. x < −5 Correct answer.
  4. x < 5

Explanation

Divide both sides by −2, which is 10 ÷ (−2) = −5. Because you divided by a negative number, the direction of the inequality symbol must reverse, so the solution is x < −5.

Question 46Algebra

Simplify by combining like terms: 4a + 3b − 2a + b

  1. 6ab
  2. 6a + 4b
  3. 2a + 4b Correct answer.
  4. 2a + 2b

Explanation

Only terms with identical variable parts can be combined: 4a − 2a = 2a and 3b + b = 4b. The simplified expression is therefore 2a + 4b, and unlike terms are never merged into one.

Question 47Algebra

Expand: −3(2x − 5)

  1. −6x − 5
  2. −6x − 15
  3. 6x − 15
  4. −6x + 15 Correct answer.

Explanation

Multiply each term inside the parentheses by −3: (−3)(2x) = −6x and (−3)(−5) = +15. The expanded form is −6x + 15, since the product of two negatives is positive.

Question 48Algebra

What is the y-intercept of the line y = 4x − 9?

  1. (0, −9) Correct answer.
  2. (0, 9)
  3. (0, 4)
  4. (4, 0)

Explanation

In slope-intercept form y = mx + b, the value b is the y-intercept, which is where the line crosses the y-axis. Here b = −9, so the intercept is the point (0, −9).

Question 49Algebra

Solve the system: y = 2x + 1 and 3x + y = 11.

  1. (3, 7)
  2. (5, 2)
  3. (2, 5) Correct answer.
  4. (1, 3)

Explanation

Substitute 2x + 1 for y in the second equation to get 3x + 2x + 1 = 11. Combining terms gives 5x = 10, so x = 2, and then y = 2(2) + 1 = 5.

Question 50Algebra

Factor completely: x² − 5x + 6

  1. (x + 2)(x + 3)
  2. (x − 1)(x − 6)
  3. (x − 2)(x − 3) Correct answer.
  4. (x + 1)(x − 6)

Explanation

You need two numbers whose product is the constant 6 and whose sum is the middle coefficient −5. The numbers −2 and −3 work because (−2)(−3) = 6 and −2 + (−3) = −5.