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Georgia Unified Math study materials
Georgia Unified Math study resources

Georgia Unified Math practice questions

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Question 1Number and Quantity

Which set of numbers is closed under subtraction?

  1. The set of positive rational numbers
  2. The set of integers Correct answer.
  3. The set of natural numbers
  4. The set of irrational numbers

Explanation

The integers are closed under subtraction because the difference of any two integers is again an integer, including when the second is larger (for example 3 − 7 = −4). The other sets fail because subtraction can leave them: natural numbers and positive rationals can produce negatives or zero, and irrationals can produce rationals such as √2 − √2 = 0.

Question 2Number and Quantity

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

  1. Every real number is either rational or irrational, and no number is both. Correct answer.
  2. Some real numbers are neither rational nor irrational.
  3. All rational numbers are also irrational numbers.
  4. Every irrational number can be written as a ratio of two integers.

Explanation

The real numbers split into two disjoint categories: numbers expressible as a ratio of integers (rational) and numbers that are not (irrational). Because the definitions are exact opposites, no real number belongs to both sets and none is left out.

Question 3Number and Quantity

A student writes 0.272727… (the block 27 repeating). Expressed as a fraction in lowest terms, this number equals which value?

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

Explanation

Let x = 0.272727…, then 100x = 27.272727…; subtracting gives 99x = 27, so x = 27/99 = 3/11. The repeating block of length two requires multiplying by 10² = 100, not by 10.

Question 4Number and Quantity

Which value is a counterexample showing that the sum of two irrational numbers is not always irrational?

  1. √2 + √2 = 2√2
  2. √3 + (−√3) = 0 Correct answer.
  3. π + 1
  4. √7 + √2

Explanation

A counterexample must be a sum of two irrationals whose result is rational. Since √3 and −√3 are both irrational and their sum is 0, which is rational, this disproves the general claim. The other sums involve irrationals but their results remain irrational.

Question 5Number and Quantity

Simplify (3 × 10⁵)(4 × 10⁻²) and write the result in scientific notation.

  1. 7 × 10³
  2. 1.2 × 10⁷
  3. 1.2 × 10⁴ Correct answer.
  4. 12 × 10³

Explanation

Multiply the coefficients: 3 × 4 = 12, and add exponents: 10⁵ · 10⁻² = 10³, giving 12 × 10³. Scientific notation requires a coefficient between 1 and 10, so rewrite 12 × 10³ as 1.2 × 10⁴.

Question 6Number and Quantity

The mass of a molecule is about 4.8 × 10⁻²³ grams. How is this written in standard decimal form?

  1. 0.000000000000000000000048 g Correct answer.
  2. 0.00000000000000000000048 g
  3. 0.0000000000000000000000048 g
  4. 48000000000000000000000 g

Explanation

A negative exponent of −23 shifts the decimal point 23 places to the left of 4.8. Writing 4.8 as 48 and moving 23 places left gives 22 zeros after the decimal point before the digits 48.

Question 7Number and Quantity

Which expression is equivalent to √50 + √18?

  1. 68
  2. 8√2 Correct answer.
  3. 4√2
  4. 2√17

Explanation

Factor each radicand into a perfect square times a remaining factor: √50 = 5√2 and √18 = 3√2. Adding like radicals gives 5√2 + 3√2 = 8√2.

Question 8Number and Quantity

Rewrite the expression 5√2 using a single radical.

  1. √50 Correct answer.
  2. √10
  3. √7
  4. √25

Explanation

A coefficient outside a radical can be moved inside by squaring it: 5√2 = √(5² · 2) = √50. Squaring is required because the coefficient multiplies the radical, not adds to it.

Question 9Number and Quantity

Evaluate 27^(2/3).

  1. 81
  2. 6
  3. 18
  4. 9 Correct answer.

Explanation

The denominator of the exponent gives a cube root and the numerator a power: 27^(2/3) = (∛27)² = 3² = 9. Taking the root first keeps the numbers small and avoids large intermediate values.

Question 10Number and Quantity

Simplify (16x⁸)^(3/4) for x > 0.

  1. 64x⁶
  2. 8x²
  3. 12x⁶
  4. 8x⁶ Correct answer.

Explanation

Apply the exponent to each factor: 16^(3/4) = (∜16)³ = 2³ = 8, and (x⁸)^(3/4) = x^(8 · 3/4) = x⁶. Combining gives 8x⁶.

Question 11Number and Quantity

Which expression is equivalent to (3 + 2i)(4 − i), where i² = −1?

  1. 14 + 5i Correct answer.
  2. 12 − 2i
  3. 14 − 5i
  4. 10 + 11i

Explanation

Expand using the distributive property: 12 − 3i + 8i − 2i². Since i² = −1, the last term becomes +2, so the real part is 12 + 2 = 14 and the imaginary part is −3i + 8i = 5i.

Question 12Number and Quantity

What is the conjugate of the complex number −7 + 4i, and what is its product with the original number?

  1. 7 − 4i, product 65
  2. 7 + 4i, product 33
  3. −7 − 4i, product 65 Correct answer.
  4. −7 − 4i, product 33

Explanation

The conjugate changes the sign of the imaginary part only, giving −7 − 4i. Their product is (−7)² + 4² = 49 + 16 = 65, since (a + bi)(a − bi) = a² + b².

Question 13Number and Quantity

Solve for real x: x² = −36.

  1. x = 18 or x = −18
  2. x = 6 or x = −6
  3. No real solution exists Correct answer.
  4. x = 36 or x = −36

Explanation

The square of any real number is nonnegative, so x² can never equal a negative number. The equation has solutions only in the complex numbers, x = ±6i, which are not real.

Question 14Number and Quantity

A quantity is measured as 6.30 × 10³ meters. How many significant figures does this measurement have?

  1. Two
  2. Three Correct answer.
  3. Four
  4. Six

Explanation

In scientific notation all digits of the coefficient are significant, and the trailing zero in 6.30 is significant because it follows a decimal point. The coefficient 6.30 therefore has three significant figures; the power of ten is not counted.

Question 15Number and Quantity

A rectangle measures 4.5 cm by 2.0 cm. Using significant-figure rules for multiplication, the area should be reported as which value?

  1. 9.0 cm²
  2. 10 cm²
  3. 9.00 cm²
  4. 9 cm² Correct answer.

Explanation

The product 4.5 × 2.0 = 9.0, but 2.0 has only two significant figures, so the answer must be rounded to two significant figures. That gives 9 cm² (two significant figures, since a trailing zero without a decimal point would not be significant).

Question 16Number and Quantity

Which statement about the units of a quantity is correct?

  1. Units may be dropped when both sides of an equation are numeric.
  2. Dimensional analysis can be used to check whether an equation is plausible. Correct answer.
  3. A ratio of two different units is always dimensionless.
  4. Adding meters to seconds gives a valid quantity.

Explanation

Dimensional analysis tracks units through an equation; if the units on each side do not match, the equation cannot be correct. This makes it a useful check before substituting numbers.

Question 17Number and Quantity

A car travels 240 kilometers in 3.0 hours. What is its average speed expressed in meters per second?

  1. About 80 m/s
  2. About 1.3 m/s
  3. About 22 m/s Correct answer.
  4. About 240 m/s

Explanation

The average speed is 240 km / 3.0 h = 80 km/h. Converting: 80 km/h × (1000 m / 1 km) × (1 h / 3600 s) ≈ 22.2 m/s.

Question 18Number and Quantity

Which expression correctly represents the product of 3.2 × 10⁻⁴ and 5 × 10⁶ in scientific notation?

  1. 8.2 × 10²
  2. 1.6 × 10⁻¹⁰
  3. 1.6 × 10³ Correct answer.
  4. 16 × 10²

Explanation

Multiply coefficients 3.2 × 5 = 16 and add exponents −4 + 6 = 2, giving 16 × 10². Since 16 is not between 1 and 10, rewrite as 1.6 × 10³.

Question 19Number and Quantity

Which of these numbers is irrational?

  1. √11 Correct answer.
  2. √16
  3. −3/7
  4. 0.75

Explanation

A square root of a non-perfect-square positive integer is irrational, and 11 is not a perfect square, so √11 is irrational. The others are a terminating decimal, a perfect square, and a ratio of integers, all rational.

Question 20Number and Quantity

Simplify (2 + i)² where i² = −1.

  1. 3 − 4i
  2. 4 + 4i
  3. 3 + 4i Correct answer.
  4. 5 + 4i

Explanation

Expand: (2 + i)(2 + i) = 4 + 4i + i². Replacing i² with −1 gives 4 + 4i − 1 = 3 + 4i.

Question 21Number and Quantity

For the complex number z = 5 − 12i, what is |z|?

  1. 7
  2. 13 Correct answer.
  3. 17
  4. 169

Explanation

The modulus is √(a² + b²) = √(5² + (−12)²) = √(25 + 144) = √169 = 13.

Question 22Number and Quantity

A student claims that √2 · √8 is irrational because it is a product of irrational factors. Which evaluation of the claim is correct?

  1. The claim is false because √2 · √8 = √16 = 4, a rational number. Correct answer.
  2. The claim is true because 8 is not a perfect square.
  3. The claim is true because both factors are irrational.
  4. The claim is false because √8 is rational.

Explanation

Multiplying radicals combines radicands: √2 · √8 = √16 = 4. Since 4 is rational, the product of two irrationals can be rational, so the student's reasoning is invalid even though each factor is irrational.

Question 23Number and Quantity

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

  1. x^(1/6)
  2. x^(0)
  3. x^(1)
  4. x^(2/3) Correct answer.

Explanation

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

Question 24Number and Quantity

A quantity is given as 3.60 × 10⁴ with three significant figures. What is the absolute uncertainty implied if the last significant digit is uncertain by ±1?

  1. ±1
  2. ±1000
  3. ±100 Correct answer.
  4. ±10

Explanation

In 3.60 × 10⁴ the last significant digit is the hundredths place of the coefficient, which corresponds to the hundreds place of the full number. An uncertainty of 1 in that place is therefore ±100.

Question 25Number and Quantity

Which statement about the product of a nonzero rational number and an irrational number is always true?

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

Explanation

Suppose r ≠ 0 is rational and y is irrational. If r·y were rational, then dividing by r (also rational) would make y rational, a contradiction. So r·y must be irrational, though it need not be an integer.

Question 26Number and Quantity

A small particle has a mass of 0.00000000000000000000000000000091 kilograms. Expressed in scientific notation, this mass is

  1. 9.1 × 10³¹ kg
  2. 91 × 10⁻³² kg
  3. 9.1 × 10⁻³¹ kg Correct answer.
  4. 9.1 × 10⁻³² kg

Explanation

To write a very small decimal in scientific notation, move the decimal point to the right until the number is between 1 and 10, then count the number of places moved. Moving the decimal 31 places to the right gives 9.1, so the exponent is −31. Thus the mass is 9.1 × 10⁻³¹ kg.

Question 27Number and Quantity

Which of these numbers is rational?

  1. π
  2. e
  3. √17
  4. √16 Correct answer.

Explanation

A rational number can be written as a ratio of two integers. Since √16 = 4, and 4 = 4/1, it is rational. The numbers √17, π, and e cannot be expressed as a simple fraction of integers.

Question 28Number and Quantity

Simplify the expression (8^(1/3) · 8^(1/6)) / 8^(1/2).

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

Explanation

Use the laws of exponents: when multiplying powers with the same base, add exponents; when dividing, subtract exponents. The numerator is 8^(1/3 + 1/6) = 8^(1/2). Dividing by 8^(1/2) gives 8^(1/2 − 1/2) = 8^0 = 1.

Question 29Number and Quantity

A quantity is measured as 7.20 × 10³ meters. How many significant figures does this measurement have?

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

Explanation

In scientific notation, all digits in the coefficient are significant. The coefficient 7.20 has three significant figures: the 7, the 2, and the trailing zero after the decimal point. The exponent does not affect the count.

Question 30Number and Quantity

For the complex number z = 3 − 4i, what is the value of z · z̄, where z̄ is the complex conjugate of z?

  1. 7
  2. 5
  3. 25 Correct answer.
  4. −7

Explanation

The complex conjugate of z = 3 − 4i is z̄ = 3 + 4i. Their product is (3 − 4i)(3 + 4i) = 3² − (4i)² = 9 − 16i². Since i² = −1, this becomes 9 + 16 = 25.

Question 31Algebra

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. In y = mx + b the coefficient m is the slope, which is -3/4.

Question 32Algebra

For the function f(x) = 2x - 7, what is f(5)?

  1. 3 Correct answer.
  2. 17
  3. 10
  4. -3

Explanation

Substitute x = 5 into the rule: 2(5) - 7 = 10 - 7 = 3. Function notation means the input replaces every occurrence of x.

Question 33Algebra

Which expression is the complete factorization of x^2 - 9?

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

Explanation

The expression is a difference of two squares, a^2 - b^2 = (a - b)(a + b), with a = x and b = 3. Expanding (x - 3)(x + 3) returns x^2 - 9.

Question 34Algebra

The system 2x + y = 8 and y = x + 2 is solved by substitution. What is the value of x?

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

Explanation

Replace y in the first equation with x + 2 to get 2x + (x + 2) = 8, so 3x = 6 and x = 2. Substituting back gives y = 4, which satisfies both equations.

Question 35Algebra

What is the value of 5^0 + 5^1?

  1. 0
  2. 6 Correct answer.
  3. 1
  4. 5

Explanation

Any nonzero base raised to the zero power equals 1, and 5^1 equals 5. Their sum is 1 + 5 = 6.

Question 36Algebra

The inequality -2x > 10 is solved for x. Which interval describes the solution?

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

Explanation

Dividing both sides by -2 reverses the inequality symbol, giving x < -5. Reversing is required whenever multiplying or dividing an inequality by a negative number.

Question 37Algebra

A line passes through the points (1, 4) and (3, 10). What is its slope?

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

Explanation

Slope equals the change in y divided by the change in x: (10 - 4)/(3 - 1) = 6/2 = 3. Order matters only in that both differences must be taken in the same direction.

Question 38Algebra

If f(x) = x^2 + 1 and g(x) = x - 3, what is f(g(5))?

  1. 5 Correct answer.
  2. 9
  3. 26
  4. 3

Explanation

Evaluate the inner function first: g(5) = 5 - 3 = 2. Then f(2) = 2^2 + 1 = 5, so the composition equals 5.

Question 39Algebra

The quadratic y = x^2 - 6x + 5 is written in vertex form. What are the coordinates of its vertex?

  1. (3, -4) Correct answer.
  2. (-3, -4)
  3. (6, 5)
  4. (3, 4)

Explanation

Complete the square: x^2 - 6x + 5 = (x - 3)^2 - 9 + 5 = (x - 3)^2 - 4. Vertex form (x - h)^2 + k gives vertex (3, -4).

Question 40Algebra

Which value of x satisfies the equation |2x - 3| = 7?

  1. 5 Correct answer.
  2. 2
  3. 7
  4. -5

Explanation

An absolute value equation splits into 2x - 3 = 7 or 2x - 3 = -7, giving x = 5 or x = -2. Only 5 appears among the choices.

Question 41Algebra

The polynomial p(x) = x^3 - 4x^2 + x + 6 has x = 2 as a root. Which expression equals p(x)?

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

Explanation

Dividing by (x - 2) leaves x^2 - 2x - 3, which factors as (x - 3)(x + 1). Multiplying confirms the constant term 6 and the leading coefficient 1.

Question 42Algebra

A geometric sequence begins 3, 6, 12, 24, ... What is the 8th term?

  1. 768
  2. 192
  3. 48
  4. 384 Correct answer.

Explanation

The common ratio is 2, and the nth term is a_1 r^(n-1) = 3 times 2^7 = 3 times 128 = 384. Counting the first term as n = 1 avoids an off-by-one error.

Question 43Algebra

The expression (x^2 - 4)/(x^2 - x - 2) is simplified for x not equal to 2 or -1. What is the result?

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

Explanation

Factor both parts: numerator (x - 2)(x + 2) and denominator (x - 2)(x + 1). Canceling the common factor (x - 2) leaves (x + 2)/(x + 1).

Question 44Algebra

For the function f(x) = 1/(x - 4), what is the domain?

  1. All real numbers greater than 4
  2. All real numbers
  3. All real numbers except x = 4 Correct answer.
  4. All real numbers except x = -4

Explanation

A rational function is undefined where its denominator equals zero. Setting x - 4 = 0 gives x = 4, so that single value is excluded from the domain.

Question 45Algebra

A line is parallel to y = (2/3)x + 5 and passes through the point (3, 1). What is the equation of the line?

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

Explanation

Parallel lines share the same slope, so m = 2/3. Using point-slope form: y - 1 = (2/3)(x - 3), which simplifies to y = (2/3)x - 1.

Question 46Algebra

The quadratic equation x^2 + kx + 9 = 0 has exactly one real solution. Which value of k is possible?

  1. 6 Correct answer.
  2. 9
  3. 3
  4. 0

Explanation

Exactly one real solution requires the discriminant b^2 - 4ac to equal zero: k^2 - 36 = 0, so k = 6 or k = -6. Only 6 is listed, and it makes the equation a perfect square (x + 3)^2.

Question 47Algebra

The function f(x) = -3(x + 1)^2 + 12 has a maximum value. What is that maximum value?

  1. 12 Correct answer.
  2. 3
  3. -1
  4. -12

Explanation

The parabola opens downward because the leading coefficient -3 is negative, so the vertex gives the maximum. In vertex form a(x - h)^2 + k, the value k = 12 is the maximum.

Question 48Algebra

If log base 2 of 8 equals y, what is y?

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

Explanation

A logarithm answers the question of what exponent on the base produces the argument. Since 2^3 = 8, log base 2 of 8 equals 3.

Question 49Algebra

The expression (2x^2 y^3)(3x^4 y)^2 is simplified completely. What is the result?

  1. 6x^10 y^6
  2. 18x^10 y^5 Correct answer.
  3. 18x^8 y^5
  4. 18x^10 y^6

Explanation

Square the second factor first: (3x^4 y)^2 = 9x^8 y^2. Multiplying by 2x^2 y^3 gives 18x^10 y^5, adding exponents with like bases.

Question 50Algebra

The function f(x) = 2x + 5 is invertible. What is f inverse of x?

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

Explanation

Swap x and y in y = 2x + 5 to get x = 2y + 5, then solve for y: y = (x - 5)/2. This undoes the original operations in reverse order.