Grade 8 Math Practice Test: Exponents, Roots and the Pythagorean Theorem
- Exponents, Roots and the Pythagorean Theorem
- 12 questions
- 15 minutes
This test covers exponent rules, square and cube roots, scientific notation, irrational numbers, the Pythagorean theorem and its converse, and the distance between two points. Tip: the hypotenuse is always the longest side of a right triangle and lies across from the right angle.
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Question 1
Which number is equal to 34 × 3−6?
Correct answer: C
When you multiply powers with the same base, add the exponents: 34 + (−6) = 3−2 = 132 = 19. A negative exponent means a reciprocal, not a negative number, so −9 is wrong.
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Question 2
What is the value of (23)2 ÷ 24?
Correct answer: A
For a power of a power, multiply the exponents: (23)2 = 26. Then 26 ÷ 24 = 26 − 4 = 22 = 4. The answer 2 comes from adding the exponents 3 + 2 instead of multiplying them.
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Question 3
What are all the solutions of x2 = 49?
Correct answer: B
Both 7 × 7 and (−7) × (−7) equal 49, so the equation has two solutions, 7 and −7. 24.5 comes from dividing 49 by 2, but squaring a number means multiplying it by itself, not by 2.
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Question 4
A cube-shaped box has a volume of 64 cubic inches. How long is each edge of the box?
Correct answer: A
The edge length is the cube root of the volume: ∛64 = 4, because 4 × 4 × 4 = 64. 8 is the square root of 64, which would give the side of a square with area 64, not the edge of a cube.
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Question 5
How is 0.00045 written in scientific notation?
Correct answer: D
Move the decimal point 4 places to the right to get 4.5, a number that is at least 1 and less than 10. Because 0.00045 is less than 1, the exponent is negative: 4.5 × 10−4. 4.5 × 104 equals 45,000.
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Question 6
Light travels about 3 × 105 kilometers per second. The Sun is about 1.5 × 108 kilometers from Earth. About how many seconds does sunlight take to reach Earth?
Correct answer: B
Time = distance ÷ speed: (1.5 ÷ 3) × 108 − 5 = 0.5 × 103 = 5 × 102, or 500 seconds, a little over 8 minutes. 5 × 103 changes 0.5 to 5 without lowering the exponent, and 4.5 × 1013 multiplies instead of dividing.
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Question 7
Which number is irrational?
Correct answer: D
18 is not a perfect square, so 18 = 4.2426… never ends or repeats and cannot be written as a fraction. 36 = 6, −27 is a fraction, and the repeating decimal 0.121212… equals 433, so those three are rational.
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Question 8
55 lies between which two consecutive whole numbers?
Correct answer: C
72 = 49 and 82 = 64, and 55 is between 49 and 64, so 55 is between 7 and 8 (it is about 7.4). 27 and 28 come from dividing 55 by 2, which is not the same as taking a square root.
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Question 9
The legs of the right triangle are 9 cm and 12 cm long.
What is the length of the hypotenuse x?
Correct answer: A
By the Pythagorean theorem, x2 = 92 + 122 = 81 + 144 = 225, so x = 225 = 15 cm. 225 cm forgets to take the square root at the end, and 21 cm simply adds the legs.
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Question 10
A 13-foot ladder leans against a vertical wall. The bottom of the ladder stands on level ground, 5 feet from the base of the wall. How high up the wall does the top of the ladder reach?
Correct answer: B
The ladder is the hypotenuse, so 52 + h2 = 132. Then h2 = 169 − 25 = 144 and h = 12 feet. About 13.9 feet (194) adds the squares, which treats the wall as the hypotenuse.
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Question 11
Which set of side lengths, in centimeters, forms a right triangle?
Correct answer: C
By the converse of the Pythagorean theorem, a triangle is a right triangle when the squares of the two shorter sides add up to the square of the longest side: 62 + 82 = 36 + 64 = 100 = 102. For 4, 5, 6 the check fails: 16 + 25 = 41, not 36.
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Question 12
What is the distance between the points (−2, 1) and (4, 9)?
Correct answer: D
The horizontal distance is 4 − (−2) = 6 and the vertical distance is 9 − 1 = 8. These are the legs of a right triangle, so the distance is 62 + 82 = 100 = 10. 14 adds the legs instead of using the Pythagorean theorem, and 68 uses 4 − 2 = 2 for the horizontal distance.
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