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OST Algebra I study materials
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OST Algebra I practice questions

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Question 1Number and Quantity

Which of the following numbers is classified as an irrational number?

  1. 0.75
  2. -6
  3. √49
  4. √2 Correct answer.

Explanation

An irrational number cannot be written as a ratio of two integers; its decimal form is non-terminating and non-repeating. The square root of 2 has a decimal expansion that never terminates or repeats, so it is irrational. The other values are all rational.

Question 2Number and Quantity

Which statement correctly describes the relationship between the rational numbers and the irrational numbers?

  1. Together they make up all real numbers, and they share no elements. Correct answer.
  2. Together they make up all real numbers, and they overlap at zero.
  3. Together they make up all natural numbers, and they share no elements.
  4. Together they make up all integers, and they share no elements.

Explanation

Every real number is either rational or irrational, never both, so the two sets are disjoint and their union is the set of real numbers. Zero is rational, so it cannot be an irrational number and the sets cannot overlap there. Integers and natural numbers are only subsets of the rationals.

Question 3Number and Quantity

A calculator displays the value of a number as 0.454554555... where each block of 4s is followed by one more 5 than before. Which classification fits this number?

  1. Integer, because it lies between 0 and 1.
  2. Natural number, because it is positive.
  3. Rational, because its digits follow a pattern.
  4. Irrational, because its decimal never terminates or repeats a fixed block. Correct answer.

Explanation

A decimal is rational only if it terminates or eventually repeats one fixed block forever. A pattern that keeps changing (adding one more 5 each time) never repeats a fixed block, so the number is irrational. Having a describable pattern does not make a decimal rational.

Question 4Number and Quantity

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

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

Explanation

Factor 50 as 25·2, then apply the product property of radicals: √50 = √25·√2 = 5√2. The largest perfect-square factor must be pulled out to reach simplest form.

Question 5Number and Quantity

Which expression is equivalent to √12 + √27?

  1. 5√3 Correct answer.
  2. 3√6
  3. 9√3
  4. √39

Explanation

Simplify each radical first: √12 = 2√3 and √27 = 3√3. Since the radicands now match, add the coefficients to get 5√3. Radicals can only be combined when the radicands are equal.

Question 6Number and Quantity

Which expression is equivalent to (3√5)(2√10)?

  1. 6√15
  2. 30√2 Correct answer.
  3. 30√5
  4. 6√50

Explanation

Multiply coefficients and radicands: (3)(2)=6 and √5·√10 = √50 = 5√2. Then 6·5√2 = 30√2. Leaving the answer as 6√50 is not simplest form because 50 still contains the perfect square 25.

Question 7Number and Quantity

Which expression is equivalent to 16^(3/4)?

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

Explanation

The denominator 4 indicates a fourth root and the numerator 3 indicates a cube. The fourth root of 16 is 2, and 2 cubed is 8. Equivalently, (16^(1/4))^3 = 2^3 = 8.

Question 8Number and Quantity

Which expression is equivalent to ∛(x^6 · y^9) for positive values of x and y?

  1. x^9·y^12
  2. x^2·y^9
  3. x^3·y^3
  4. x^2·y^3 Correct answer.

Explanation

Rewrite the cube root as the exponent 1/3 and multiply exponents: x^(6·1/3) = x^2 and y^(9·1/3) = y^3. The result is x^2·y^3. Dividing each exponent by the index 3 gives the same conclusion.

Question 9Number and Quantity

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

  1. 16x^7
  2. 2x^12
  3. 8x^12
  4. 16x^12 Correct answer.

Explanation

The exponent 4 applies to every factor inside the parentheses. Raise the coefficient: 2^4 = 16, and multiply the exponent on x: 3·4 = 12, giving 16x^12.

Question 10Number and Quantity

Which expression is equivalent to (x^5)/(x^8) for x ≠ 0?

  1. -x^3
  2. 1/x^13
  3. x^3
  4. 1/x^3 Correct answer.

Explanation

When dividing powers with the same base, subtract the exponents: 5 - 8 = -3, so the result is x^(-3). A negative exponent means the reciprocal, so x^(-3) = 1/x^3.

Question 11Number and Quantity

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

  1. -16
  2. -8
  3. 16
  4. 1/16 Correct answer.

Explanation

A negative exponent means take the reciprocal of the base raised to the positive exponent: 4^(-2) = 1/(4^2) = 1/16. The negative sign in the exponent never makes the result negative.

Question 12Number and Quantity

Which expression is equivalent to (5√3)^2?

  1. 15
  2. 75 Correct answer.
  3. 30
  4. 25√3

Explanation

Square each factor: 5^2 = 25 and (√3)^2 = 3. Multiplying gives 25·3 = 75. Squaring a radical expression removes the radical entirely.

Question 13Number and Quantity

A square has an area of 72 square centimeters. Which expression gives the exact side length in simplest radical form?

  1. 8√3 cm
  2. 6√2 cm Correct answer.
  3. 2√18 cm
  4. 36√2 cm

Explanation

The side length is √72. Since 72 = 36·2, we get √72 = 6√2. This is the simplest form because 72 has no larger perfect-square factor than 36.

Question 14Number and Quantity

Which value is the rationalized form of 6/√3?

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

Explanation

Multiply numerator and denominator by √3: (6√3)/(√3·√3) = (6√3)/3 = 2√3. Rationalizing removes the radical from the denominator without changing the value.

Question 15Number and Quantity

Which expression is equivalent to (2 + √5)(3 - √5)?

  1. 6 - 5
  2. 1 + √5 Correct answer.
  3. 11 + 5√5
  4. 11 - √5

Explanation

Use the distributive property: 2·3 = 6, 2·(-√5) = -2√5, √5·3 = 3√5, and √5·(-√5) = -5. Combining gives 1 + √5. The radical terms combine because -2√5 + 3√5 = √5.

Question 16Number and Quantity

Which expression is equivalent to √(x^2) when x is any real number?

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

Explanation

The principal square root is never negative, so √(x^2) equals the absolute value of x, which returns x when x is nonnegative and -x when x is negative. Writing just x fails for negative inputs.

Question 17Number and Quantity

The expression 7^(2/3) can be rewritten in radical form. Which choice is correct?

  1. ∛(7^2) Correct answer.
  2. ∛(7^3)
  3. (∛7)^3
  4. √(7^3)

Explanation

In a fractional exponent m/n, the denominator n is the index of the radical and the numerator m is the power of the radicand. So 7^(2/3) = ∛(7^2). The order of rooting and raising to a power may be swapped, but the index must be 3 and the power 2.

Question 18Number and Quantity

Which expression is equivalent to (x^(1/2))(x^(1/3)) for x > 0?

  1. x^(5/6) Correct answer.
  2. x^(2/3)
  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). Exponents are added, not multiplied, in a product.

Question 19Number and Quantity

Which expression is equivalent to (27x^9)^(2/3)?

  1. 9x^6 Correct answer.
  2. 9x^9
  3. 18x^6
  4. 3x^6

Explanation

Apply the exponent 2/3 to each factor. For 27: 27^(2/3) = (∛27)^2 = 3^2 = 9. For x^9: multiply exponents, 9·(2/3) = 6, giving x^6. The result is 9x^6.

Question 20Number and Quantity

Which of these numbers is a rational number?

  1. √7
  2. √2 + 1
  3. 0.333... repeating Correct answer.
  4. π

Explanation

A repeating decimal such as 0.333... equals 1/3, a ratio of integers, so it is rational. Numbers such as π and √7 have non-terminating, non-repeating decimals and cannot be written as a fraction of integers.

Question 21Number and Quantity

A student claims that 0.101001000100001... (with one more zero between each pair of 1s) is rational because the pattern is easy to describe. Which evaluation of the claim is correct?

  1. The claim is wrong, because the number terminates.
  2. The claim is wrong, because the decimal never settles into a fixed repeating block. Correct answer.
  3. The claim is correct, because any decimal with a rule is rational.
  4. The claim is correct, because the number is less than 1.

Explanation

A decimal represents a rational number only if it terminates or eventually repeats one fixed block of digits forever. Growing runs of zeros mean no fixed block ever repeats, so the number is irrational. Being describable by a rule, or being less than 1, does not make a decimal rational.

Question 22Number and Quantity

Which expression is equivalent to √8 · √2?

  1. √10
  2. 4 Correct answer.
  3. 2√2
  4. 16

Explanation

The product property of radicals gives √8·√2 = √16 = 4. Multiplying the radicands first is valid because both are nonnegative.

Question 23Number and Quantity

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

  1. 1/∜(x^3) Correct answer.
  2. -∜(x^3)
  3. ∜(x^3)
  4. -1/∜(x^3)

Explanation

A negative exponent signals a reciprocal, so x^(-3/4) = 1/x^(3/4). Rewriting the positive fractional exponent in radical form gives 1/∜(x^3). The negative sign in the exponent never makes the value negative.

Question 24Number and Quantity

The expression ∜(81x^12) simplifies to a monomial for x > 0. Which choice gives the simplified result?

  1. 3x^4
  2. 81x^3
  3. 9x^3
  4. 3x^3 Correct answer.

Explanation

The fourth root of 81 is 3, because 3^4 = 81. For the variable, divide the exponent by the index: 12/4 = 3, giving x^3. The simplified monomial is 3x^3.

Question 25Number and Quantity

Which statement about the closure of the rational numbers is correct?

  1. The sum of two rational numbers is always rational, and their product is always rational. Correct answer.
  2. The sum of two rational numbers may be irrational, but their product is always rational.
  3. The sum of two rational numbers is always rational, but their product may be irrational.
  4. Neither the sum nor the product of two rational numbers is guaranteed to be rational.

Explanation

If a/b and c/d are rational, their sum (ad + bc)/(bd) and product (ac)/(bd) are both ratios of integers with nonzero denominators, so both are rational. The rational numbers are closed under addition and multiplication.

Question 26Number and Quantity

Which expression is equivalent to 5√12 − 2√27?

  1. 10√3
  2. 16√3
  3. 4√3 Correct answer.
  4. 3√3

Explanation

Simplify each radical first: √12 = 2√3, so 5√12 = 10√3; and √27 = 3√3, so 2√27 = 6√3. Subtracting like radicals gives 10√3 − 6√3 = 4√3.

Question 27Number and Quantity

A number is written as 3.6 × 10^(-4) in scientific notation. Which decimal is equivalent?

  1. 0.0000036
  2. 0.000036
  3. 0.00036 Correct answer.
  4. 36,000

Explanation

A negative exponent of −4 moves the decimal point 4 places to the left from 3.6. Moving left four places gives 0.00036, so 3.6 × 10^(-4) = 0.00036.

Question 28Number and Quantity

Which expression is equivalent to (8x^6)^(1/3) for all real x?

  1. 24x^2
  2. 2x^2 Correct answer.
  3. 2x^3
  4. 8x^2

Explanation

The exponent 1/3 applies to each factor: 8^(1/3) = 2 and (x^6)^(1/3) = x^2 by the power rule. The product is 2x^2.

Question 29Number and Quantity

Which statement correctly describes the sum of a rational number and an irrational number?

  1. The sum is always irrational. Correct answer.
  2. The sum is always an integer.
  3. The sum is always rational.
  4. The sum may be either rational or irrational.

Explanation

If a rational plus an irrational gave a rational sum, subtracting the rational would leave the irrational as a difference of two rationals, which is impossible. So the sum of a rational and an irrational is always irrational.

Question 30Number and Quantity

Which expression is equivalent to √50 + √18 − √8?

  1. 2√10
  2. 8√2
  3. 6√2 Correct answer.
  4. 4√2

Explanation

Simplify each term: √50 = 5√2, √18 = 3√2, and √8 = 2√2. Combining like radicals gives 5√2 + 3√2 − 2√2 = 6√2.

Question 31Number and Quantity

Which expression is equivalent to (9/16)^(-1/2)?

  1. 4/3 Correct answer.
  2. 16/9
  3. 3/4
  4. -3/4

Explanation

A negative exponent takes the reciprocal, so (9/16)^(-1/2) = (16/9)^(1/2). The square root of 16/9 is 4/3, since (4/3)^2 = 16/9.

Question 32Number and Quantity

Which expression is equivalent to (2√3 − 1)(2√3 + 1)?

  1. 12 − 4√3
  2. 12 + 4√3
  3. 13
  4. 11 Correct answer.

Explanation

The factors form a difference of squares: (2√3)^2 − 1^2. Since (2√3)^2 = 4 · 3 = 12, the value is 12 − 1 = 11.

Question 33Number and Quantity

A square has an area of 108 square inches. What is the perimeter of the square in simplest radical form?

  1. 24√3 inches Correct answer.
  2. 6√3 inches
  3. 36√3 inches
  4. 12√3 inches

Explanation

The side length is √108 = 6√3, since 108 = 36 · 3. The perimeter of a square is four times the side, so 4 · 6√3 = 24√3 inches.

Question 34Number and Quantity

Which value is equivalent to 1/(3 + √5) in rationalized form?

  1. (3 − √5)/14
  2. (3 − √5)/4 Correct answer.
  3. (3 + √5)/4
  4. (√5 − 3)/4

Explanation

Multiply numerator and denominator by the conjugate 3 − √5. The denominator becomes 9 − 5 = 4, and the numerator becomes 3 − √5, giving (3 − √5)/4.

Question 35Number and Quantity

Which expression is equivalent to (x^(2/3))^(9/4) for x > 0?

  1. x^(18/13)
  2. x^(11/12)
  3. x^(3/2) Correct answer.
  4. x^(8/27)

Explanation

When a power is raised to a power, multiply the exponents: (2/3)(9/4) = 18/12 = 3/2. So the expression equals x^(3/2).

Question 36Number and Quantity

Which expression is equivalent to ∛(54x^5) for x > 0?

  1. 2x·∛(3x^2)
  2. 3x·∛(2x^2) Correct answer.
  3. 9x·∛(2x^2)
  4. 3x^2·∛(2x)

Explanation

Factor 54 as 27 · 2 and x^5 as x^3 · x^2. Then ∛(27x^3) = 3x pulls out front, leaving ∛(2x^2) inside, so the result is 3x·∛(2x^2).

Question 37Number and Quantity

Which expression is equivalent to √(x^7) · √(x^5) for x ≥ 0?

  1. x^6 Correct answer.
  2. x^12
  3. x^5·√x
  4. x^6·√x

Explanation

Multiply under one radical: √(x^7 · x^5) = √(x^12). Since x ≥ 0, √(x^12) = x^6, because (x^6)^2 = x^12.

Question 38Number and Quantity

Which expression is equivalent to (x^(1/2) + 3)(x^(1/2) − 3) for x ≥ 0?

  1. x − 9 Correct answer.
  2. x − 6
  3. x + 9
  4. x − 6x^(1/2) + 9

Explanation

The product matches the difference-of-squares pattern (a + b)(a − b) = a^2 − b^2 with a = x^(1/2) and b = 3. Since (x^(1/2))^2 = x, the result is x − 9.

Question 39Number and Quantity

A student rewrites √(a^2 · b^4) as a·b^2 for all real numbers a and b. Which statement about this rewrite is correct?

  1. It is valid for all real a and b, since even powers always give positive roots.
  2. It is never valid, because radicals cannot be split across products.
  3. It is valid only when b ≥ 0, because b^4 has an odd exponent.
  4. It is valid only when a ≥ 0, because √(a^2) equals |a| for all real a. Correct answer.

Explanation

For any real a, √(a^2) = |a|, not a, so replacing √(a^2) with a is only correct when a is nonnegative. The factor b^4 is a perfect square with √(b^4) = b^2 for all real b, so the restriction concerns a alone.

Question 40Number and Quantity

The number N = 0.272727... repeats the block 27 forever. Which expression shows N as a rational number in lowest terms?

  1. 27/100
  2. 11/3
  3. 27/99
  4. 3/11 Correct answer.

Explanation

Let N = 0.272727... Then 100N = 27.272727..., and subtracting gives 99N = 27, so N = 27/99 = 3/11. A repeating decimal always represents a rational number.

Question 41Algebra

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

  1. 6x - 12
  2. 11x - 4
  3. 6x + 5x - 12
  4. 11x - 12 Correct answer.

Explanation

Distribute the 3 across the parentheses to get 6x - 12, then combine like terms with 5x. The x-terms add to 11x while the constant stays -12, giving 11x - 12.

Question 42Algebra

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

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

Explanation

Substitute x = -2: 2(-2)^2 - 3(-2) + 1. Since (-2)^2 = 4, this becomes 8 + 6 + 1 = 15. Note that squaring a negative gives a positive result.

Question 43Algebra

Solve the equation 5x - 7 = 3x + 9.

  1. x = 1
  2. x = 8 Correct answer.
  3. x = 16
  4. x = 2

Explanation

Subtract 3x from both sides to get 2x - 7 = 9, then add 7 to both sides to get 2x = 16. Dividing by 2 gives x = 8.

Question 44Algebra

Which expression represents the slope of the line passing through the points (2, 5) and (6, 13)?

  1. (13 + 5)/(6 + 2)
  2. (6 - 2)/(13 - 5)
  3. (13 - 5)/(6 - 2) Correct answer.
  4. (6 - 2)/(5 - 13)

Explanation

The slope formula is the change in y divided by the change in x: (y2 - y1)/(x2 - x1). Using the two points gives (13 - 5)/(6 - 2), which simplifies to 2.

Question 45Algebra

Simplify (x^3)(x^4).

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

Explanation

When multiplying powers with the same base, add the exponents: x^3 times x^4 equals x^(3+4) = x^7. The base stays the same.

Question 46Algebra

Which equation is in slope-intercept form with slope -2 and y-intercept 3?

  1. 2x + y = 3
  2. y - 3 = -2x
  3. y = 3x - 2
  4. y = -2x + 3 Correct answer.

Explanation

Slope-intercept form is y = mx + b, where m is the slope and b is the y-intercept. Substituting m = -2 and b = 3 gives y = -2x + 3.

Question 47Algebra

Factor the expression x^2 - 9.

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

Explanation

This is a difference of squares: a^2 - b^2 = (a - b)(a + b). Here a = x and b = 3, so x^2 - 9 factors as (x - 3)(x + 3).

Question 48Algebra

Solve the inequality -3x < 12.

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

Explanation

Divide both sides by -3, and remember to reverse the inequality symbol when dividing by a negative number. This gives x > -4.

Question 49Algebra

What is the slope of a line perpendicular to a line with slope 3/4?

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

Explanation

Perpendicular lines have slopes that are negative reciprocals of each other. The negative reciprocal of 3/4 is -4/3.

Question 50Algebra

Solve the system: y = 2x + 1 and y = -x + 7.

  1. (4, 9)
  2. (1, 3)
  3. (2, 5) Correct answer.
  4. (3, 7)

Explanation

Set the two expressions for y equal: 2x + 1 = -x + 7. Solving gives 3x = 6, so x = 2, and substituting back gives y = 5. The solution is (2, 5).