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NY Regents Algebra I study materials
NY Regents Algebra I study resources

NY Regents Algebra I practice questions

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Question 1Number & Quantity

Which statement correctly classifies the number 0?

  1. 0 is a whole number, an integer, and a rational number. Correct answer.
  2. 0 is irrational because it cannot be written as a fraction.
  3. 0 is rational but not a whole number.
  4. 0 is a natural number but not an integer.

Explanation

Zero belongs to the whole numbers, the integers, and the rationals, since it equals 0/1, a ratio of integers. It is not irrational, and the natural numbers usually begin at 1.

Question 2Number & Quantity

A calculator displays 4.5 × 10^5 for a measurement. Which value is shown in standard decimal form?

  1. 450,000 Correct answer.
  2. 0.000045
  3. 4,500,000
  4. 45,000

Explanation

The exponent 5 on the base 10 means multiply 4.5 by five factors of ten, that is by 100,000. So 4.5 × 10^5 = 450,000.

Question 3Number & Quantity

Which expression is equivalent to 5√2 + 3√2 − √2?

  1. 8√2
  2. 7√2 Correct answer.
  3. 7√6
  4. 9√2

Explanation

Radicals with the same radicand behave like like terms: add and subtract the coefficients and keep √2. The coefficients are 5 + 3 − 1 = 7, so the result is 7√2.

Question 4Number & Quantity

Which list orders the numbers from least to greatest: −3/4, −0.5, 0.2, −0.75?

  1. 0.2, −0.5, −3/4, −0.75
  2. −0.5, −0.75, −3/4, 0.2
  3. −3/4, −0.5, 0.2, −0.75
  4. −0.75, −3/4, −0.5, 0.2 Correct answer.

Explanation

Rewrite −3/4 as −0.75 so all values share a decimal form. On a number line the negatives lie left of zero, and −0.75 is the farthest left, followed by −0.5, then the positive 0.2.

Question 5Number & Quantity

Which of these numbers is irrational?

  1. √17 Correct answer.
  2. 0.125
  3. √16
  4. 22/7

Explanation

A number is irrational when it cannot be written as a ratio of integers; its decimal expansion never terminates or repeats. Since 17 is not a perfect square, √17 is irrational, unlike the other three choices.

Question 6Number & Quantity

What is the value of 2^3 · 2^4?

  1. 2^1
  2. 2^12
  3. 2^7 Correct answer.
  4. 4^7

Explanation

When multiplying powers with the same base, keep the base and add the exponents: 2^3 · 2^4 = 2^(3+4) = 2^7. The base does not change and the exponents are not multiplied.

Question 7Number & Quantity

A length is given as 6.2 × 10^(-3) meters. Which decimal represents this length?

  1. 6200
  2. 0.00062
  3. 0.0062 Correct answer.
  4. 0.062

Explanation

A negative exponent moves the decimal point to the left. Moving it three places in 6.2 gives 0.0062, which is the value of 6.2 × 10^(-3).

Question 8Number & Quantity

Which expression shows 72 written as a product of a perfect-square factor and another integer, so it can simplify √72?

  1. 24 · 3
  2. 9 · 8
  3. 4 · 18
  4. 36 · 2 Correct answer.

Explanation

To simplify a square root, factor out the largest perfect square. Since 72 = 36 · 2 and 36 is a perfect square, √72 = 6√2, the simplest form.

Question 9Number & Quantity

Which property justifies the statement 7(x + 4) = 7x + 28?

  1. Commutative property of addition
  2. Additive identity property
  3. Distributive property Correct answer.
  4. Associative property of multiplication

Explanation

The distributive property states a(b + c) = ab + ac, multiplying the outside factor by each term inside. Here 7 is distributed over x and 4 to give 7x + 28.

Question 10Number & Quantity

Simplify (3x^2 y)(−2x^4 y^3).

  1. 6x^6 y^4
  2. −6x^8 y^3
  3. −5x^6 y^4
  4. −6x^6 y^4 Correct answer.

Explanation

Multiply the coefficients: 3 · (−2) = −6. For each variable, add exponents of like bases: x^(2+4) = x^6 and y^(1+3) = y^4, giving −6x^6 y^4.

Question 11Number & Quantity

Which value of x makes the expression √(x − 5) a real number?

  1. x = 0
  2. x = 5 Correct answer.
  3. x = 3
  4. x = −2

Explanation

A square root is a real number only when the radicand is nonnegative, so x − 5 ≥ 0, meaning x ≥ 5. Of the choices, only x = 5 satisfies this condition.

Question 12Number & Quantity

A calculator shows 3.6 × 10^4 multiplied by 2.0 × 10^2. What is the product in scientific notation?

  1. 7.2 × 10^8
  2. 5.6 × 10^6
  3. 7.2 × 10^6 Correct answer.
  4. 7.2 × 10^2

Explanation

Multiply the coefficients and add the exponents: (3.6)(2.0) = 7.2 and 10^(4+2) = 10^6. The product is 7.2 × 10^6, already in proper scientific notation.

Question 13Number & Quantity

Simplify (2^5 · 2^3) / 2^4.

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

Explanation

In the numerator, add exponents: 2^5 · 2^3 = 2^8. Then divide by subtracting exponents: 2^8 / 2^4 = 2^(8−4) = 2^4, which equals 16.

Question 14Number & Quantity

Which expression is equivalent to 4√3 + √12?

  1. 5√3
  2. 4√15
  3. 16√3
  4. 6√3 Correct answer.

Explanation

Simplify √12 = √(4 · 3) = 2√3. Then 4√3 + 2√3 = 6√3, since the radicands now match and only the coefficients add.

Question 15Number & Quantity

The mass of a particle is about 0.00000042 grams. Which expression gives this mass in scientific notation?

  1. 42 × 10^(-8)
  2. 4.2 × 10^7
  3. 4.2 × 10^(-6)
  4. 4.2 × 10^(-7) Correct answer.

Explanation

Scientific notation writes a number as a × 10^n with 1 ≤ a < 10. Moving the decimal 7 places right in 0.00000042 gives 4.2, so the value is 4.2 × 10^(-7).

Question 16Number & Quantity

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

  1. The sum is always an integer.
  2. The sum is always rational.
  3. The sum is sometimes rational and sometimes irrational.
  4. The sum is always irrational. Correct answer.

Explanation

If a rational plus an irrational were rational, subtracting the rational would make the irrational rational, a contradiction. So the sum of a rational and an irrational is always irrational, as with 2 + √3.

Question 17Number & Quantity

Simplify √50 − √18.

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

Explanation

Simplify each radical: √50 = 5√2 and √18 = 3√2. Subtracting coefficients gives 5√2 − 3√2 = 2√2.

Question 18Number & Quantity

Evaluate (3 × 10^8)(4 × 10^(-3)) and express the result in scientific notation.

  1. 7 × 10^5
  2. 1.2 × 10^5
  3. 1.2 × 10^6 Correct answer.
  4. 12 × 10^5

Explanation

Multiply coefficients: 3 · 4 = 12. Add exponents: 10^(8 + (−3)) = 10^5, giving 12 × 10^5. Rewrite 12 as 1.2 × 10^1 so the result is 1.2 × 10^6.

Question 19Number & Quantity

Which expression is equivalent to (x^3)^4 · x^(-2)?

  1. x^24
  2. x^14
  3. x^10 Correct answer.
  4. x^5

Explanation

A power raised to a power multiplies exponents: (x^3)^4 = x^12. Then multiply by x^(-2) by adding exponents: x^(12 + (−2)) = x^10.

Question 20Number & Quantity

The number 0.272727… repeats the block 27 forever. Which fraction equals this decimal?

  1. 27/99 Correct answer.
  2. 272/1000
  3. 27/10
  4. 27/100

Explanation

A repeating decimal with a two-digit block equals the block over as many 9s as there are digits. So 0.272727… = 27/99, which simplifies to 3/11.

Question 21Number & Quantity

Which expression is equivalent to 5/(2 − √3) after rationalizing the denominator?

  1. 5(2 + √3) Correct answer.
  2. 5(2 − √3)
  3. 10 + 5√3 over 7
  4. (2 + √3)/5

Explanation

Multiply numerator and denominator by the conjugate 2 + √3. The denominator becomes (2)^2 − (√3)^2 = 4 − 3 = 1, so the expression simplifies to 5(2 + √3).

Question 22Number & Quantity

Simplify (8x^6 y^3)^(2/3) for positive x and y.

  1. 4x^9 y^2
  2. 4x^4 y^3
  3. 16x^4 y^2
  4. 4x^4 y^2 Correct answer.

Explanation

Apply the exponent to each factor: 8^(2/3) = 4, x^(6 · 2/3) = x^4, and y^(3 · 2/3) = y^2. The simplified expression is 4x^4 y^2.

Question 23Number & Quantity

The expression (2 × 10^a)(5 × 10^b) equals 1 × 10^9, where a and b are integers. Which equation must relate a and b?

  1. a · b = 9
  2. a − b = 8
  3. a + b = 9
  4. a + b = 8 Correct answer.

Explanation

Multiply coefficients: 2 · 5 = 10 = 1 × 10^1. The product is 10^(1 + a + b), which must equal 10^9, so 1 + a + b = 9 and therefore a + b = 8.

Question 24Number & Quantity

Which expression equals 1/(√7 − √5) written with a rational denominator?

  1. √7 + √5
  2. (√7 + √5)/12
  3. (√7 + √5)/2 Correct answer.
  4. (√7 − √5)/2

Explanation

Multiply by the conjugate √7 + √5 over itself. The denominator becomes 7 − 5 = 2, so the expression equals (√7 + √5)/2.

Question 25Number & Quantity

A number is written as 6.0 × 10^n, where n is an integer. For which set of values of n is the value of the number greater than 1 but less than 6.0 × 10^2?

  1. n = 0 and n = 1 Correct answer.
  2. n = 1 and n = 2
  3. n = 2 and n = 3
  4. n = −1 and n = 0

Explanation

With n = 0 the value is 6.0, and with n = 1 it is 60, both between 1 and 600. When n = 2 the value is 600, which is not less than 6.0 × 10^2, and negative n gives values below 1.

Question 26Number & Quantity

Which expression is equivalent to √48 in simplest radical form?

  1. 3√4
  2. 4√3 Correct answer.
  3. 8√6
  4. 2√12

Explanation

Factor 48 as 16 · 3, and since 16 is a perfect square, √48 = √16 · √3 = 4√3. Simplest radical form requires the largest perfect-square factor to be removed from the radicand.

Question 27Number & Quantity

A number is classified as irrational. Which statement must be true of that number?

  1. Its decimal expansion eventually repeats one block forever.
  2. Its decimal expansion terminates after finitely many digits.
  3. It can be written as a ratio of two integers.
  4. Its decimal expansion neither terminates nor repeats. Correct answer.

Explanation

An irrational number cannot be expressed as a ratio of integers, which is equivalent to saying its decimal expansion is infinite and non-repeating. Terminating and repeating decimals both represent rational numbers.

Question 28Number & Quantity

Evaluate (2 × 10^5)^3 and write the result in scientific notation.

  1. 2 × 10^15
  2. 6 × 10^15
  3. 8 × 10^8
  4. 8 × 10^15 Correct answer.

Explanation

Apply the power to each factor: 2^3 = 8 and (10^5)^3 = 10^15 by multiplying exponents. The product 8 × 10^15 already has a coefficient between 1 and 10, so it is in scientific notation.

Question 29Number & Quantity

Which expression is equivalent to (5 + 2i) + (3 − 7i), where i^2 = −1?

  1. 2 − 5i
  2. 8 − 5i Correct answer.
  3. 15 − 14i
  4. 8 + 9i

Explanation

Add real parts together and imaginary parts together: 5 + 3 = 8 and 2i − 7i = −5i. The sum is 8 − 5i.

Question 30Number & Quantity

Which expression is equivalent to √18 + √8 − √32?

  1. √2 Correct answer.
  2. −√2
  3. 3√2
  4. 5√2

Explanation

Simplify each radical: √18 = 3√2, √8 = 2√2, and √32 = 4√2. Combining like radicals gives 3√2 + 2√2 − 4√2 = √2.

Question 31Number & Quantity

Which expression is equivalent to (4x^2)^(1/2) · (9x^4)^(1/2) for x > 0?

  1. 36x^3
  2. 6x^2
  3. 6x^3 Correct answer.
  4. 13x^3

Explanation

Each factor is a square root: (4x^2)^(1/2) = 2x and (9x^4)^(1/2) = 3x^2 for positive x. Multiplying gives 2x · 3x^2 = 6x^3.

Question 32Number & Quantity

Which property of exponents justifies rewriting 1/x^(−3) as x^3 for x ≠ 0?

  1. A negative exponent indicates the reciprocal of the base raised to the opposite exponent. Correct answer.
  2. A zero exponent makes any nonzero base equal to 1.
  3. The product of powers with the same base adds the exponents.
  4. A power raised to a power multiplies the exponents.

Explanation

By definition, x^(−n) = 1/x^n, so 1/x^(−3) = 1/(1/x^3) = x^3. The rule converts a negative exponent into a reciprocal with a positive exponent.

Question 33Number & Quantity

A calculator displays a result as 5.4E−7. Which standard decimal represents this value?

  1. 0.000000054
  2. 0.00000054 Correct answer.
  3. 54000000
  4. 0.0000000540

Explanation

The display 5.4E−7 means 5.4 × 10^(−7). Moving the decimal point 7 places left gives 0.00000054.

Question 34Number & Quantity

Which expression is equivalent to (6 − 2i)(3 + 4i), where i^2 = −1?

  1. 18 − 8i
  2. 26 − 18i
  3. 10 + 18i
  4. 26 + 18i Correct answer.

Explanation

Distribute: 6·3 + 6·4i − 2i·3 − 2i·4i = 18 + 24i − 6i − 8i^2. Since i^2 = −1, replace −8i^2 with +8, giving 26 + 18i.

Question 35Number & Quantity

Which expression is equivalent to 3/(√5 + 1) after rationalizing the denominator?

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

Explanation

Multiply numerator and denominator by the conjugate √5 − 1. The denominator becomes (√5)^2 − 1^2 = 4, and the numerator becomes 3(√5 − 1), so the result is 3(√5 − 1)/4.

Question 36Number & Quantity

Which expression is equivalent to (16x^8)^(3/4) for x > 0?

  1. 8x^6 Correct answer.
  2. 12x^6
  3. 64x^6
  4. 8x^2

Explanation

Rewrite as (16x^8)^(3/4) = (16^(1/4))^3 · (x^8)^(3/4). Since 16^(1/4) = 2, the coefficient is 2^3 = 8, and multiplying exponents gives x^6.

Question 37Number & Quantity

Which expression is equivalent to (2√6)(3√10) in simplest radical form?

  1. 12√60
  2. 12√15 Correct answer.
  3. 6√60
  4. 6√15

Explanation

Multiply coefficients and radicands separately: 2·3 = 6 and √6·√10 = √60 = 2√15. Then 6 · 2√15 = 12√15, which is fully simplified.

Question 38Number & Quantity

A rational number r and an irrational number s are given. Which conclusion about the product r · s is valid when r ≠ 0?

  1. The product must be irrational. Correct answer.
  2. The product must equal zero.
  3. The product may be either rational or irrational.
  4. The product must be rational.

Explanation

If r · s were rational, dividing by the nonzero rational r would make s rational, contradicting the given. Therefore the product of a nonzero rational and an irrational number is always irrational.

Question 39Number & Quantity

The expression (4 × 10^m)(2.5 × 10^n) is written in scientific notation as a × 10^k, where 1 ≤ a < 10. Given that m + n = 6, what are the values of a and k?

  1. a = 10, k = 6
  2. a = 1, k = 7 Correct answer.
  3. a = 10, k = 7
  4. a = 1, k = 6

Explanation

Multiply the coefficients: 4 · 2.5 = 10, and combine powers to get 10^6, so the product is 10 × 10^6 = 10^7. Scientific notation requires a coefficient between 1 and 10, so rewrite as 1 × 10^7, giving a = 1 and k = 7.

Question 40Number & Quantity

For positive real numbers a and b, which expression is equivalent to √(a^3 b^5) · √(a b)?

  1. a b^3
  2. a^2 b^3 Correct answer.
  3. a^3 b^3
  4. a^2 b^2

Explanation

Combine the radicands: √(a^3 b^5 · a b) = √(a^4 b^6). Since a and b are positive, taking square roots gives a^2 b^3, halving each exponent.

Question 41Algebra

Simplify the expression 3(2x + 4) - 5x.

  1. x + 4
  2. 6x - 1
  3. 11x + 12
  4. x + 12 Correct answer.

Explanation

Distribute the 3 to get 6x + 12, then combine like terms with -5x: 6x - 5x = x, so the result is x + 12. Combining like terms only joins terms with identical variable parts.

Question 42Algebra

What is the value of 2x² - 3x + 1 when x = -2?

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

Explanation

Substitute x = -2: 2(-2)² - 3(-2) + 1 = 2(4) + 6 + 1 = 8 + 6 + 1 = 15. Squaring a negative gives a positive, and subtracting a negative product adds.

Question 43Algebra

Solve the equation 5x - 7 = 2x + 8.

  1. x = 1
  2. x = 3
  3. x = 15
  4. x = 5 Correct answer.

Explanation

Subtract 2x from both sides to get 3x - 7 = 8, then add 7 to get 3x = 15, so x = 5. Each step keeps the equation balanced by applying the same operation to both sides.

Question 44Algebra

Which expression represents the factored form of x² - 9?

  1. (x + 3)(x + 3)
  2. (x - 3)(x + 3) Correct answer.
  3. (x - 3)(x - 3)
  4. (x - 9)(x + 1)

Explanation

x² - 9 is a difference of two squares, a² - b² = (a - b)(a + b), with a = x and b = 3. Expanding (x - 3)(x + 3) gives x² + 3x - 3x - 9 = x² - 9, confirming the factorization.

Question 45Algebra

A line passes through the points (2, 5) and (6, 13). What is the slope of the line?

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

Explanation

Slope equals the change in y divided by the change in x: (13 - 5)/(6 - 2) = 8/4 = 2. The slope measures vertical change per unit of horizontal change.

Question 46Algebra

What is the y-intercept of the graph of the equation y = -4x + 7?

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

Explanation

In slope-intercept form y = mx + b, the value b is the y-intercept, so here b = 7. The graph crosses the y-axis at the point (0, 7).

Question 47Algebra

Simplify the expression (2x³)(5x⁴).

  1. 7x⁷
  2. 10x⁷ Correct answer.
  3. 10x¹²
  4. 7x¹²

Explanation

Multiply the coefficients (2·5 = 10) and add the exponents when multiplying powers with the same base (x³·x⁴ = x⁷). The product is 10x⁷.

Question 48Algebra

If f(x) = 3x - 2, what is f(4)?

  1. 6
  2. 12
  3. 14
  4. 10 Correct answer.

Explanation

Substitute x = 4 into the function rule: f(4) = 3(4) - 2 = 12 - 2 = 10. Function notation f(4) means the output when the input is 4.

Question 49Algebra

Solve the inequality -3x + 4 < 19.

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

Explanation

Subtract 4 from both sides to get -3x < 15, then divide by -3 and reverse the inequality sign to get x > -5. Dividing or multiplying an inequality by a negative number flips the direction of the inequality.

Question 50Algebra

What are the solutions to x² - 5x + 6 = 0?

  1. x = 1 and x = 6
  2. x = -1 and x = -6
  3. x = -2 and x = -3
  4. x = 2 and x = 3 Correct answer.

Explanation

Factor the quadratic as (x - 2)(x - 3) = 0, since -2 and -3 multiply to 6 and add to -5. Setting each factor equal to zero gives x = 2 and x = 3.