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NYS Math practice questions

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Question 1Number and Quantity

Which of the following numbers is irrational?

  1. √7 Correct answer.
  2. 0.75
  3. √16
  4. 22/7

Explanation

An irrational number cannot be written as a ratio of two integers, and its decimal expansion is non-terminating and non-repeating. The square root of 7 is not a perfect square, so it is irrational, whereas √16 equals 4, 0.75 terminates, and 22/7 is a fraction of integers.

Question 2Number and Quantity

What is the value of (2³)(2⁴) expressed as a single power of 2?

  1. 2¹²
  2. 2¹
  3. 4⁷
  4. 2⁷ Correct answer.

Explanation

When multiplying powers with the same base, add the exponents: 2³·2⁴ = 2^(3+4) = 2⁷. The bases stay the same and only the exponents combine.

Question 3Number and Quantity

Which expression is equivalent to √50 in simplest radical form?

  1. 25√2
  2. 2√5
  3. 10√5
  4. 5√2 Correct answer.

Explanation

Factor 50 as 25·2, and since 25 is a perfect square, √50 = √25·√2 = 5√2. Simplifying radicals means extracting the largest perfect-square factor.

Question 4Number and Quantity

The number 0.333... (the digit 3 repeating forever) belongs to which set?

  1. Natural numbers
  2. Integers
  3. Irrational numbers only
  4. Rational numbers Correct answer.

Explanation

A repeating decimal can always be written as a fraction of integers; here 0.333... = 1/3. Because it is a ratio of integers, the number is rational, though it is neither an integer nor a natural number.

Question 5Number and Quantity

What is the value of 5⁰?

  1. 5
  2. 1 Correct answer.
  3. Undefined
  4. 0

Explanation

Any nonzero base raised to the zero power equals 1. This follows from the quotient rule: 5²/5² = 5^(2−2) = 5⁰, and any nonzero number divided by itself is 1.

Question 6Number and Quantity

Which list orders the numbers from least to greatest: −3/4, −0.5, −1, 0.2?

  1. −1, −3/4, −0.5, 0.2 Correct answer.
  2. −0.5, −3/4, −1, 0.2
  3. 0.2, −0.5, −3/4, −1
  4. −1, −0.5, −3/4, 0.2

Explanation

On a number line, more negative values are smaller. Since −1 < −0.75 < −0.5 < 0.2, the correct order is −1, −3/4, −0.5, 0.2.

Question 7Number and Quantity

Which of the following is a rational number?

  1. 0.1010010001... (non-repeating pattern)
  2. √2
  3. π
  4. −8/2 Correct answer.

Explanation

A rational number can be written as a ratio of two integers. The expression −8/2 equals −4, an integer, so it is rational; π, √2, and non-repeating infinite decimals are irrational.

Question 8Number and Quantity

Simplify the expression (3²)⁴.

  1. 3⁸ Correct answer.
  2. 3¹⁶
  3. 9⁸
  4. 3⁶

Explanation

When a power is raised to a power, multiply the exponents: (3²)⁴ = 3^(2×4) = 3⁸. The base remains 3.

Question 9Number and Quantity

Which value is equivalent to 2⁻³?

  1. −8
  2. 1/6
  3. 1/8 Correct answer.
  4. −6

Explanation

A negative exponent indicates a reciprocal: 2⁻³ = 1/2³ = 1/8. The negative sign on the exponent does not make the result negative.

Question 10Number and Quantity

Simplify √12 + √27.

  1. √39
  2. 6√3
  3. 5√3 Correct answer.
  4. 3√13

Explanation

Simplify each radical first: √12 = 2√3 and √27 = 3√3. Since these are like radicals, add the coefficients to get 5√3.

Question 11Number and Quantity

What is the product (3 + √5)(3 − √5)?

  1. 4 Correct answer.
  2. 14
  3. 9 + 5√5
  4. 9 − 5√5

Explanation

This is a difference of squares: (3)² − (√5)² = 9 − 5 = 4. The cross terms +3√5 and −3√5 cancel.

Question 12Number and Quantity

Which expression is equivalent to √(x⁶) for x > 0?

  1. x⁴
  2. x¹²
  3. x³ Correct answer.
  4. x⁹

Explanation

The square root of a power divides the exponent by 2: √(x⁶) = x^(6/2) = x³. Equivalently, (x³)² = x⁶.

Question 13Number and Quantity

Write 0.000045 in scientific notation.

  1. 4.5 × 10⁻⁶
  2. 4.5 × 10⁵
  3. 45 × 10⁻⁶
  4. 4.5 × 10⁻⁵ Correct answer.

Explanation

Move the decimal point 5 places to the right to get 4.5, so the exponent is −5: 4.5 × 10⁻⁵. Scientific notation requires a coefficient between 1 and 10.

Question 14Number and Quantity

Simplify (2 × 10⁸) ÷ (4 × 10³) in scientific notation.

  1. 2 × 10⁵
  2. 5 × 10¹¹
  3. 5 × 10⁵ Correct answer.
  4. 0.5 × 10⁵

Explanation

Divide the coefficients: 2 ÷ 4 = 0.5, and subtract the exponents: 10^(8−3) = 10⁵, giving 0.5 × 10⁵. Convert to proper form by moving the decimal: 5 × 10⁴... wait, 0.5 × 10⁵ = 5 × 10⁴, so the coefficient must be adjusted.

Question 15Number and Quantity

Which statement is true about the sum of a rational number and an irrational number?

  1. It is always rational.
  2. It is always irrational. Correct answer.
  3. It could be either rational or irrational.
  4. It is always an integer.

Explanation

If a rational plus an irrational were rational, subtracting the rational would leave a rational difference, contradicting that the second number is irrational. Therefore the sum must be irrational.

Question 16Number and Quantity

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

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

Explanation

A fractional exponent m/n means the nth root raised to the mth power. The fourth root of 16 is 2, and 2³ = 8, so 16^(3/4) = 8.

Question 17Number and Quantity

Simplify the expression (8x⁶)^(2/3) for x > 0.

  1. 4x⁹
  2. 2x⁴
  3. 16x⁴
  4. 4x⁴ Correct answer.

Explanation

Apply the exponent 2/3 to each factor: 8^(2/3) = (∛8)² = 4, and (x⁶)^(2/3) = x^(6·2/3) = x⁴. The result is 4x⁴.

Question 18Number and Quantity

Which expression is equivalent to (x^(1/2))(x^(1/3))?

  1. x^(2/3)
  2. x^(5/6) Correct answer.
  3. x^(2/5)
  4. x^(1/6)

Explanation

When multiplying powers with the same base, add the exponents: 1/2 + 1/3 = 3/6 + 2/6 = 5/6. The product is x^(5/6).

Question 19Number and Quantity

A number is rational if and only if its decimal expansion does which of the following?

  1. Continues forever without repeating
  2. Terminates or eventually repeats a block of digits Correct answer.
  3. Is greater than zero
  4. Contains only the digits 0 and 1

Explanation

A rational number's decimal expansion either terminates or eventually repeats a fixed block of digits. Conversely, any such decimal can be converted back into a fraction of integers.

Question 20Number and Quantity

Which value is the smallest?

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

Explanation

Compare the exponents first: 10⁻⁴ is the smallest power of ten here, and 3 × 10⁻⁴ = 0.0003 is less than 2 × 10⁻³ = 0.002. Thus 3 × 10⁻⁴ is the smallest value.

Question 21Number and Quantity

What is the value of 27^(−2/3)?

  1. 9
  2. 1/9 Correct answer.
  3. −1/9
  4. −9

Explanation

Rewrite the negative exponent as a reciprocal: 27^(−2/3) = 1/27^(2/3). Since ∛27 = 3 and 3² = 9, the result is 1/9.

Question 22Number and Quantity

Rationalize the denominator of 6/(√5 − √2).

  1. 6(√5 + √2)
  2. 2(√5 + √2) Correct answer.
  3. 2(√5 − √2)
  4. (√5 + √2)/2

Explanation

Multiply numerator and denominator by the conjugate √5 + √2. The denominator becomes 5 − 2 = 3, so the expression is 6(√5 + √2)/3 = 2(√5 + √2).

Question 23Number and Quantity

If a and b are positive integers and √a · √b = 12, which pair (a, b) is impossible?

  1. (9, 16)
  2. (2, 72)
  3. (4, 36)
  4. (3, 60) Correct answer.

Explanation

The equation requires ab = 144. Checking each pair: 9·16 = 144, 4·36 = 144, 2·72 = 144, but 3·60 = 180, so √3·√60 = √180 ≠ 12.

Question 24Number and Quantity

Simplify (x^(2/3) · x^(1/6)) / x^(1/2) for x > 0.

  1. x^(1/3) Correct answer.
  2. x^(1/6)
  3. x^(1/2)
  4. x^(5/6)

Explanation

In the numerator, add exponents: 2/3 + 1/6 = 4/6 + 1/6 = 5/6. Then subtract the denominator's exponent: 5/6 − 1/2 = 5/6 − 3/6 = 2/6 = 1/3, giving x^(1/3).

Question 25Number and Quantity

The number N = 2^10 · 5^8 is written in standard decimal form. How many digits does N have?

  1. 18
  2. 8
  3. 9 Correct answer.
  4. 10

Explanation

Rewrite N = 2²·(2⁸·5⁸) = 4·10⁸. Since 4 × 10⁸ = 400,000,000, which has 9 digits, N has 9 digits.

Question 26Number and Quantity

Which of the following numbers is an integer?

  1. −3/4
  2. −7 Correct answer.
  3. √2
  4. 0.25

Explanation

An integer is a whole number that can be positive, negative, or zero, with no fractional or decimal part. −7 is a whole number, while −3/4, √2, and 0.25 are not integers because they are fractional, irrational, or decimal.

Question 27Number and Quantity

Which statement correctly describes the set of irrational numbers?

  1. They include all square roots of positive integers.
  2. They are always negative.
  3. Their decimal expansions neither terminate nor repeat. Correct answer.
  4. They can be written as a ratio of two integers.

Explanation

Irrational numbers cannot be expressed as a fraction of two integers, and their decimal expansions are infinite and non-repeating. For example, π and √2 have non-terminating, non-repeating decimals. The other statements mischaracterize irrational numbers.

Question 28Number and Quantity

Evaluate the expression (5^2 · 5^3) / 5^4.

  1. 5 Correct answer.
  2. 25
  3. 1/5
  4. 125

Explanation

When multiplying powers with the same base, add the exponents: 5^2 · 5^3 = 5^(2+3) = 5^5. Dividing by 5^4 subtracts exponents: 5^(5−4) = 5^1 = 5. So the expression simplifies to 5.

Question 29Number and Quantity

Which expression is equivalent to √50 − √18?

  1. √32
  2. 2
  3. 2√2 Correct answer.
  4. 8√2

Explanation

Simplify each radical: √50 = √(25·2) = 5√2, and √18 = √(9·2) = 3√2. Subtracting gives 5√2 − 3√2 = 2√2. The other options do not simplify to this value.

Question 30Number and Quantity

If a is a positive rational number and b is a positive irrational number, which statement about a·b must be true?

  1. The product is irrational. Correct answer.
  2. The product is zero.
  3. The product is rational.
  4. The product could be either rational or irrational.

Explanation

The product of a nonzero rational number and an irrational number is always irrational. If a·b were rational, then dividing by a (which is nonzero) would make b rational, contradicting that b is irrational. Therefore a·b must be irrational.

Question 31Number and Quantity

What is the value of 64^(2/3)?

  1. 4
  2. 8
  3. 32
  4. 16 Correct answer.

Explanation

The exponent 2/3 means take the cube root of 64 and then square the result. The cube root of 64 is 4, and 4^2 = 16. Therefore 64^(2/3) = 16.

Question 32Number and Quantity

Which expression is equivalent to (16x^8)^(1/4) for x > 0?

  1. 2x^4
  2. 4x^2
  3. 4x^4
  4. 2x^2 Correct answer.

Explanation

Apply the exponent 1/4 to each factor: 16^(1/4) = 2, and (x^8)^(1/4) = x^(8·1/4) = x^2. Thus the expression simplifies to 2x^2. The other options incorrectly evaluate the coefficient or the exponent.

Question 33Number and Quantity

Which value is equal to 5^(−2)?

  1. −25
  2. 25
  3. 1/25 Correct answer.
  4. −1/25

Explanation

A negative exponent indicates the reciprocal of the base raised to the positive exponent: 5^(−2) = 1/(5^2) = 1/25. The negative sign in the exponent does not make the result negative.

Question 34Number and Quantity

The number 0.00000000072 is written in scientific notation as which expression?

  1. 72 × 10^(−11)
  2. 7.2 × 10^(−10) Correct answer.
  3. 7.2 × 10^(−11)
  4. 7.2 × 10^(−9)

Explanation

To write 0.00000000072 in scientific notation, move the decimal point 10 places to the right to get 7.2. Because the original number is less than 1, the exponent is −10. Thus 0.00000000072 = 7.2 × 10^(−10).

Question 35Number and Quantity

What is the value of (3 × 10^5) · (2 × 10^(−8)) expressed in scientific notation?

  1. 5 × 10^(−3)
  2. 6 × 10^(−2)
  3. 6 × 10^(−3) Correct answer.
  4. 6 × 10^(−4)

Explanation

Multiply the coefficients: 3 · 2 = 6. Add the exponents: 5 + (−8) = −3. The result is 6 × 10^(−3), which is already in scientific notation because 6 is between 1 and 10.

Question 36Number and Quantity

Which expression is equivalent to (x^(3/4))^(2/3) for x > 0?

  1. x^(1/12)
  2. x^(2)
  3. x^(9/8)
  4. x^(1/2) Correct answer.

Explanation

When raising a power to a power, multiply the exponents: (3/4) · (2/3) = 6/12 = 1/2. Therefore (x^(3/4))^(2/3) = x^(1/2).

Question 37Number and Quantity

Which expression is equivalent to √(50x^3) for x > 0?

  1. 5x√(2x^2)
  2. 5x√(2x) Correct answer.
  3. 5x^2√(2x)
  4. 25x√(2x)

Explanation

Factor the radicand into perfect squares: 50x^3 = 25 · 2 · x^2 · x. The square root of 25x^2 is 5x, leaving √(2x). Thus √(50x^3) = 5x√(2x).

Question 38Number and Quantity

If n is a positive integer, which expression is equivalent to (√[3]{8})^n?

  1. 8^(n/3)
  2. 8^n
  3. 2^(n/3)
  4. 2^n Correct answer.

Explanation

The cube root of 8 is 2, because 2^3 = 8. Raising 2 to the power n gives 2^n. Therefore (√[3]{8})^n = 2^n.

Question 39Number and Quantity

The number N = 2^5 · 5^3 is written in standard decimal form. How many digits does N have?

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

Explanation

Combine factors to form powers of 10: 2^5 · 5^3 = 2^2 · (2^3 · 5^3) = 4 · 10^3 = 4000. The number 4000 has four digits: the leading 4 and three zeros.

Question 40Number and Quantity

Which expression is equivalent to (√2 + √3)^2?

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

Explanation

Expand the square: (√2 + √3)^2 = (√2)^2 + 2(√2)(√3) + (√3)^2 = 2 + 2√6 + 3 = 5 + 2√6. The cross term 2√6 is essential and cannot be omitted.

Question 41Algebra

What is the slope of the line whose equation is 3x + 4y = 12?

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

Explanation

Solve for y to reach slope-intercept form: 4y = -3x + 12, so y = -3/4 x + 3. The coefficient of x is the slope, which is -3/4.

Question 42Algebra

Which value of x satisfies the equation 5(x - 2) = 3x + 4?

  1. 9
  2. 1
  3. 3
  4. 7 Correct answer.

Explanation

Distribute to get 5x - 10 = 3x + 4, then subtract 3x from both sides: 2x - 10 = 4. Adding 10 gives 2x = 14, so x = 7.

Question 43Algebra

The function f is defined by f(x) = 2x² - 3x + 1. What is f(-2)?

  1. 3
  2. -1
  3. 11
  4. 15 Correct answer.

Explanation

Substitute -2 carefully: 2(-2)² - 3(-2) + 1 = 2(4) + 6 + 1. This equals 8 + 6 + 1 = 15.

Question 44Algebra

A system consists of y = 2x + 1 and y = -x + 7. What is the x-coordinate of its solution?

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

Explanation

Set the expressions equal: 2x + 1 = -x + 7. Adding x and subtracting 1 gives 3x = 6, so x = 2.

Question 45Algebra

Which expression is equivalent to (x³)(x⁵)?

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

Explanation

When multiplying powers with the same base, add the exponents: x³ · x⁵ = x^(3+5) = x⁸. The base stays the same and the coefficients are not combined.

Question 46Algebra

The graph of y = x² - 4x + 3 crosses the x-axis at which pair of values?

  1. x = -1 and x = -3
  2. x = -1 and x = 3
  3. x = 1 and x = 3 Correct answer.
  4. x = 1 and x = -3

Explanation

Factor the quadratic as (x - 1)(x - 3) = 0. The zero-product property gives x = 1 or x = 3, the x-intercepts of the parabola.

Question 47Algebra

What is the slope of any line parallel to the line passing through (1, 4) and (5, 12)?

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

Explanation

Slope equals the change in y divided by the change in x: (12 - 4)/(5 - 1) = 8/4 = 2. Parallel lines share the same slope, so the answer is 2.

Question 48Algebra

If g(x) = x - 5, what is g⁻¹(9)?

  1. 9
  2. -14
  3. 14 Correct answer.
  4. 4

Explanation

To invert, solve y = x - 5 for x, giving x = y + 5, so g⁻¹(x) = x + 5. Then g⁻¹(9) = 9 + 5 = 14.

Question 49Algebra

Simplify (2x²y)(3xy⁴).

  1. 6x²y⁴
  2. 6x³y⁴
  3. 5x³y⁵
  4. 6x³y⁵ Correct answer.

Explanation

Multiply coefficients: 2 · 3 = 6. For each variable add the exponents: x² · x¹ = x³ and y¹ · y⁴ = y⁵, giving 6x³y⁵.

Question 50Algebra

What is the solution set of the inequality -2x + 5 > 11?

  1. x > -3
  2. x < 3
  3. x > 3
  4. x < -3 Correct answer.

Explanation

Subtract 5 from both sides to get -2x > 6. Dividing by -2 reverses the inequality sign, so x < -3.