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Grade 8 math review pack

30 mixed questions from every topic in Grade 8, with an answer key and explanations. Great for the start of the year, before a test, or over the holidays.

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Grade 8 Math Review Pack

Name: ____________________Date: ____________Score: ____ / 30
  1. Squares and Square Roots
    $43^2$ = ?
    Answer: ______________________
  2. Square Roots and Radicals
    Simplify $\dfrac{6}{5 + 2\sqrt{3}} + \dfrac{1}{5 - 2\sqrt{3}}$.
    A. $\dfrac{7}{13}$B. $\dfrac{25 - 14\sqrt{3}}{13}$C. $\dfrac{35 + 10\sqrt{3}}{13}$D. $\dfrac{35 - 10\sqrt{3}}{13}$
  3. Simplify $5\sqrt{6} \times 5\sqrt{6}$.
    A. 300B. $50\sqrt{3}$C. 150D. 60
  4. $\dfrac{\sqrt{108} - \sqrt{108}}{\sqrt{3}}$ = ?
    Answer: ______________________
  5. Simplify $\sqrt{99}$.
    A. $-3\sqrt{11}$B. $3\sqrt{11}$C. $4\sqrt{11}$D. $6\sqrt{11}$
  6. Exponents
    Simplify $\dfrac{(-5x^{2}y^{4})(2x^{4}y^{5})}{6x^{3}y^{5}}$. Use positive exponents.
    A. $-\dfrac{5x^{2}y^{4}}{3}$B. $\dfrac{5x^{3}y^{4}}{3}$C. $-\dfrac{5x^{3}y^{4}}{3}$D. $-\dfrac{5x^{3}y^{6}}{3}$
  7. Simplify $\dfrac{(3x^{4}y^{-3})^{3}}{6x^{-3}y^{5}}$. Use positive exponents only.
    A. $\dfrac{9x^{14}}{2y^{14}}$B. $\dfrac{9y^{14}}{2x^{15}}$C. $\dfrac{9x^{15}}{2y^{12}}$D. $\dfrac{9x^{15}}{2y^{14}}$
  8. $-4^{2}$ = ?
    Answer: ______________________
  9. Solve: $3^{x} = 9^{x - 3}$
    Answer: ______________________
  10. Algebra: Expanding
    Expand $(4x - 5)(x + 1)$.
    A. $4x^{2} + x - 5$B. $4x^{2} + 9x - 5$C. $4x^{2} - x - 5$D. $4x^{2} - x - 6$
  11. Expand and simplify $3x(x + 2) - 4(x^{2} - x - 1)$.
    A. $-x^{2} + 7x + 4$B. $x^{2} - 10x - 4$C. $7x^{2} + 2x - 4$D. $-x^{2} + 10x + 4$
  12. Systems of Equations
    A jar has 27 coins, all dimes (10¢) and quarters (25¢). The total value is $\$4.20$. How many quarters are there?
    Answer: ______________________
  13. Solve the system: $x + y - 2z = -8$, $x + 3y + 2z = 12$, $-x - y - 3z = -22$. Give (x, y, z).
    A. $(-2, 6, 6)$B. $(-6, -2, 6)$C. $(6, -2, 6)$D. $(6, 6, -2)$
  14. Solve by substitution: $x - 3y = -11$ and $5x - 3y = 5$. Give (x, y).
    A. $(5, 5)$B. $(4, 4)$C. $(4, 5)$D. $(-4, 5)$
  15. How many solutions does this system have? $5x + 3y = 5$ and $-10x - 6y = -8$
    A. Exactly oneB. No solutionC. Infinitely many
  16. Geometry Theorems
    From an outside point P, two tangents touch the circle at A and B. PA = (4x − 2) cm and PB = (3x + 4) cm. How long is PA? (cm)
    Answer: ______________________
  17. What is each interior angle of a regular 72-sided polygon? (degrees)
    Answer: ______________________
  18. In triangle ABC, AB = AC and ∠A = 36°. Side BC is extended past C to D. ∠ACD = (4x + 20)°. Find x.
    Answer: ______________________
  19. The horizontal lines are parallel. The marked angles are 6x+25 and 4x+55. Find x.
    Answer: ______________________
  20. Probability
    In a survey: Grade 7 — 6 play a sport, 26 do not. Grade 8 — 28 play a sport, 14 do not. A Grade 8 student is picked at random. What is the probability they play a sport? Write it as a fraction.
    Answer: ______________________
  21. 8 fair coins are tossed. What is the probability of exactly 3 heads? Write it as a fraction.
    Answer: ______________________
  22. In a game you roll a die. A 6 wins $\$2$, a 1, 2 or 3 wins $\$2$, and a 4 or 5 wins nothing. What is the expected winning per roll, in dollars? Write it as a fraction.
    Answer: ______________________
  23. A bag has 4 red and 3 blue marbles. Two marbles are drawn with replacement (the first is put back). What is the probability that both are red? Write it as a fraction.
    Answer: ______________________
  24. Quick Review
    Write the exact area of an equilateral triangle with sides 25 cm.
    A. $\dfrac{625}{4}\sqrt{3}\text{ cm}^2$B. $156.25\text{ cm}^2$C. $312.5\sqrt{3}\text{ cm}^2$D. $625\sqrt{3}\text{ cm}^2$
  25. A motorboat goes 54 km downstream in 3 h and 56 km upstream in 4 h. What is the boat's speed in still water? (km/h)
    Answer: ______________________
  26. A person 177 cm tall casts a 228 cm shadow. At the same time a building casts a 1,900 cm shadow. How tall is the building in metres, to one decimal place?
    Answer: ______________________
  27. A triangular field has sides 25 m, 39 m and 16 m. Sod costs 5 dollars per m². How much does it cost to cover the field? (dollars)
    Answer: ______________________
  28. Mixed Challenges
    x + y = 11 and xy = 16. What is $x^2 + y^2$?
    Answer: ______________________
  29. Solve $x^2 = 8$ for the positive value of x. Write it in simplest radical form.
    A. 4B. $\sqrt{4}$C. $4\sqrt{2}$D. $2\sqrt{2}$
  30. A rectangle has sides of (3x + 3) cm and (2x + 5) cm. Its perimeter is 86 cm. What is its area? (cm²)
    Answer: ______________________

Answer key

  1. 1849 — Split it: (40 + 3)² = 1600 + 2 × 40 × 3 + 9 = 1,849.
  2. D ($\dfrac{35 - 10\sqrt{3}}{13}$) — The denominators are conjugates, so the common denominator is $(5 + 2\sqrt{3})(5 - 2\sqrt{3}) = 25 - 12 = 13$. Numerator: $6(5 - 2\sqrt{3}) + 1(5 + 2\sqrt{3}) = 35 - 10\sqrt{3}$. Answer: $\dfrac{35 - 10\sqrt{3}}{13}$.
  3. C (150) — Multiply the outside numbers and the inside numbers: 5 × 5 = 25 and $\sqrt{6 \times 6} = \sqrt{36}$. So the answer is $25\sqrt{36}$ = 150.
  4. 0 — Divide each term by $\sqrt{3}$: $\sqrt{36} - \sqrt{36} = 6 - 6 = 0$.
  5. B ($3\sqrt{11}$) — Find the biggest perfect square factor: 99 = 9 × 11. $\sqrt{9} = 3$, so $\sqrt{99} = 3\sqrt{11} = 3\sqrt{11}$.
  6. C ($-\dfrac{5x^{3}y^{4}}{3}$) — Top: $(−5x^{2}y^{4})(2x^{4}y^{5}) = -10x^{6}y^{9}$. Divide by $6x^{3}y^{5}$: numbers -10 ÷ 6 = -5/3, subtract exponents. Negative exponents move to the bottom: $-\dfrac{5x^{3}y^{4}}{3}$.
  7. D ($\dfrac{9x^{15}}{2y^{14}}$) — 1. Power of a power: $(3x^{4}y^{-3})^{3} = 27x^{12}y^{-9}$. 2. Divide by $6x^{-3}y^{5}$: subtract exponents → $\dfrac{9}{2}x^{15}y^{-14}$. Positive exponents: $\dfrac{9x^{15}}{2y^{14}}$.
  8. -16 — Only 4 is raised to the power (the minus sign is outside): $-(4^{2}) = -16$.
  9. 6 — Write both as powers of 3: $3^{1(x)} = 3^{2(x - 3)}$. Set the exponents equal: 1(x) = 2(x - 3) → 1x + 0 = 2x − 6 → x = 6.
  10. C ($4x^{2} - x - 5$) — FOIL: $4x \cdot x = 4x^{2}$; middle $4x + -5x = -x$; last $(−5) \times 1 = -5$. Answer: $4x^{2} - x - 5$.
  11. D ($-x^{2} + 10x + 4$) — First: $3x(x + 2) = 3x^{2} + 6x$. Second: $-4(x^{2} - x - 1) = -4x^{2} + 4x + 4$ (the minus multiplies every term!). Collect like terms: $-x^{2} + 10x + 4$.
  12. 10 — d + q = 27 and 10d + 25q = 420. Multiply the first by 10 and subtract: 15q = 150, q = 10.
  13. C ($(6, -2, 6)$) — 1. Eliminate x using equations 1 and 2: $y + 2z = 10$. 2. Eliminate x using equations 1 and 3: $z = 6$. 3. Eliminate y from these two: $z = 6$, so z = 6. 4. Substitute back to get y = −2, then x = 6. Check in all three equations.
  14. C ($(4, 5)$) — Rearrange the first: $x = -11 + 3y$. Substitute: $5(−11 + 3y) - 3y = 5$ → $-55 + 12y = 5$ → y = 5. Then x = -11 − (−3)(5) = 4.
  15. B (No solution) — Multiply the first equation by -2: $-10x - 6y = -10$. The left sides match but the right sides don't, so the lines are parallel: no solution.
  16. 22 — Tangents from the same outside point are equal: 4x − 2 = 3x + 4 → 1x = 6 → x = 6. PA = 4(6) − 2 = 22 cm.
  17. 175 — Sum = (72 − 2) × 180° = 12600°. Divide by 72 equal angles: 175°. (Or: each exterior angle is 360° ÷ 72 = 5°, and 180° − 5° = 175°.)
  18. 22 — ITT: AB = AC, so ∠B = ∠C = (180 − 36) ÷ 2 = 72°. Angles on a straight line: ∠ACD = 180 − 72 = 108° (or EAT: 36 + 72). So 4x + 20 = 108, 4x = 88, x = 22.
  19. 15 — Alternate angles are equal (Z pattern): 6x+25 = 4x+55. Solve: 2x = 30, x = 15.
  20. 2/3 — Only look at Grade 8: 42 students, 28 play a sport. P(sport | Grade 8) = 28/42 = 2/3. ("Given Grade 8" shrinks the sample.)
  21. 7/32 — Each of the 256 outcomes is equally likely. Ways to choose which 3 coins are heads: C(8, 3) = 56. Probability = 56/256 = 7/32.
  22. 4/3 — Expected value = sum of (prize × probability) = 2 × 1/6 + 2 × 3/6 + 0 × 2/6 = 4/3 dollars.
  23. 16/49 — P(red, red) = 4/7 × 4/7 = 16/49.
  24. A ($\dfrac{625}{4}\sqrt{3}\text{ cm}^2$) — Height = (√3 ⁄ 2) × 25. Area = ½ × 25 × (√3 ⁄ 2) × 25 = (625 ⁄ 4)√3 = 625/4 √3 cm².
  25. 16 — Downstream speed: 54 ÷ 3 = 18. Upstream speed: 56 ÷ 4 = 14. Boat speed = average of the two = (18 + 14) ÷ 2 = 16 km/h.
  26. 14.8 — Similar triangles: height ÷ shadow is equal. Building = 177 × 1,900 ÷ 228 = 1475 cm ≈ 14.8 m.
  27. 600 — s = 40. Area = √(40 × 15 × 1 × 24) = 120 m². Cost = 120 × 5 = 600 dollars.
  28. 89 — $(x + y)^2 = x^2 + 2xy + y^2$, so $x^2 + y^2 = 11^2 - 2(16) = 89$.
  29. D ($2\sqrt{2}$) — $x = \sqrt{8} = \sqrt{4 \times 2} = 2\sqrt{2}$.
  30. 456 — Perimeter: 2(5x + 8) = 86 → x = 7. The sides are 24 cm and 19 cm. Area = 456 cm². (Or expand: $(3x + 3)(2x + 5) = 6x^{2} + 21x + 15$ at x = 7.)

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