CBSE Class 8 Mathematics Cubes And Cube Roots Worksheet Set 05

Choose the correct option (Question No. 1 - 4)

Question 1. The cube root of -64 is
(a) 2
(b) -2
(c) -4
(d) 4
Answer: (c) -4
We know that \((-4) \times (-4) \times (-4) = -64\). Therefore, the cube root of -64 is -4.
In simple words: Multiplying -4 three times gives -64.

Exam Tip: The cube root of a negative number is always negative.

Question 2. 36 is multiplied by \(x\) to get a perfect cube. The value of \(x\) is
(a) 32
(b) 2
(c) 6
(d) 4
Answer: (c) 6
Prime factorization of \(36 = 2^2 \times 3^2\). To make it a perfect cube, each prime factor must have an exponent that is a multiple of 3. We need one more 2 and one more 3, so \(x = 2 \times 3 = 6\).
In simple words: We need triplets for prime factorization, so multiply by missing factors.

Exam Tip: Always find prime factors and complete groups of three.

Question 3. Which of the following is a square number and also a perfect cube?
(a) 144
(b) 125
(c) 16
(d) 64
Answer: (d) 64
\(64 = 8^2\) (hence a square number) and \(64 = 4^3\) (hence a perfect cube).
In simple words: 64 can be expressed as both \(8^2\) and \(4^3\).

Exam Tip: Numbers which are sixth powers of integers (like \(2^6 = 64\)) are both squares and cubes.

Question 4. The cube root of 1.331 is
(a) 1.1
(b) 0.011
(c) 0.11
(d) 1.01
Answer: (a) 1.1
\(1.1 \times 1.1 \times 1.1 = 1.331\). Therefore, the cube root is 1.1.
In simple words: Cube of 11 is 1331, so cube of 1.1 is 1.331.

Exam Tip: Count decimal places; the cube root will have one-third the number of decimal places.

Question 5. Fill in the blanks
(a) The cube of an odd natural number is odd.
(b) The cube of a negative integer is negative.
(c) Cube of a natural number along with one zero ends with three zeros.
Answer: (a) odd, (b) negative, (c) three zeros
In simple words: Properties of cubes preserve signs and multiply the count of trailing zeros by 3.

Exam Tip: Remember standard cube properties for quick filling.

Question 6. Find the cube of
(a) 0.8
(b) 4.3
(c) -5.1
(d) \(-\frac{1}{8}\)
(e) \(-\frac{3}{2}\)
Answer:
(a) \(0.8^3 = 0.8 \times 0.8 \times 0.8 = 0.512\)
(b) \(4.3^3 = 4.3 \times 4.3 \times 4.3 = 79.507\)
(c) \((-5.1)^3 = (-5.1) \times (-5.1) \times (-5.1) = -132.651\)
(d) \(\left(-\frac{1}{8}\right)^3 = \frac{-1}{512}\)
(e) \(\left(-\frac{3}{2}\right)^3 = -\frac{27}{8}\)
In simple words: Multiply the number by itself three times.

Exam Tip: Be careful with decimal placement and negative signs.

Question 7. Find by using Prime factorization whether the following are perfect cubes?
(a) 13284
(b) 91125
(c) 10648
(d) 17576
(e) 39304
Answer:
(a) \(13284 = 2^2 \times 3^3 \times 123\). Since all prime factors do not form triplets, 13284 is not a perfect cube.
(b) \(91125 = 3^6 \times 5^3 = (3^2 \times 5)^3 = 45^3\). Since all prime factors form triplets, 91125 is a perfect cube.
(c) \(10648 = 2^3 \times 11^3 = (2 \times 11)^3 = 22^3\). Since all prime factors form triplets, 10648 is a perfect cube.
(d) \(17576 = 2^3 \times 13^3 = (2 \times 13)^3 = 26^3\). Since all prime factors form triplets, 17576 is a perfect cube.
(e) \(39304 = 2^3 \times 17^3 = (2 \times 17)^3 = 34^3\). Since all prime factors form triplets, 39304 is a perfect cube.
In simple words: Group prime factors in triplets of three. If none are left over, it is a perfect cube.

Exam Tip: Always group factors in triplets to check for perfect cubes.

Question 8. Find the least number by which the following must be multiplied to make them perfect cubes.
(a) 1323
(b) 5400
(c) 14400
(d) 2560
Answer:
(a) \(1323 = 3^3 \times 7^2\). To make it a cube, we need one more 7. Least number = 7.
(b) \(5400 = 2^3 \times 3^3 \times 5^2\). To make it a cube, we need one more 5. Least number = 5.
(c) \(14400 = 2^6 \times 3^2 \times 5^2\). We need one 3 and one 5 to complete triplets. Least number = \(3 \times 5 =\) 15.
(d) \(2560 = 2^9 \times 5\). We need two more 5s. Least number = \(5 \times 5 =\) 25.
In simple words: Find the missing factors needed to complete groups of three.

Exam Tip: Multiply the deficient prime factors to find the required multiplier.

Question 9. Find the least number by which the following must be divided to make them perfect cubes.
(a) 53240
(b) 1024
(c) 2187
(d) 5400
Answer:
(a) \(53240 = 2^3 \times 11^3 \times 5\). Dividing by 5 removes the extra factor and leaves a perfect cube.
(b) \(1024 = 2^{10} = 2^3 \times 2^3 \times 2^3 \times 2\). Dividing by 2 leaves \(2^9\), which is a perfect cube.
(c) \(2187 = 3^7 = 3^3 \times 3^3 \times 3\). Dividing by 3 leaves \(3^6\), which is a perfect cube.
(d) \(5400 = 2^3 \times 3^3 \times 5^2\). Dividing by \(5^2 =\) 25 leaves \(2^3 \times 3^3\), which is a perfect cube.
In simple words: Divide out the extra prime factors that do not form a complete triplet.

Exam Tip: The divisors are the leftover prime factors that do not form triplets.

Question 10. Find the cube root of
(a) 1331
(b) 3375
(c) 15625
(d) 35937
(e) 42875
(f) 13824
(g) \(18\frac{26}{27}\)
(h) \(-\frac{343}{729}\)
(i) -0.001331
(j) 0.002197
Answer:
(a) \(\sqrt[3]{1331} = 11\)
(b) \(\sqrt[3]{3375} = 15\)
(c) \(\sqrt[3]{15625} = 25\)
(d) \(\sqrt[3]{35937} = 33\)
(e) \(\sqrt[3]{42875} = 35\)
(f) \(\sqrt[3]{13824} = 24\)
(g) \(18\frac{26}{27} = \frac{512}{27}\). Cube root = \(\frac{8}{3}\)
(h) \(\sqrt[3]{-\frac{343}{729}} = -\frac{7}{9}\)
(i) \(\sqrt[3]{-0.001331} = -0.11\)
(j) \(\sqrt[3]{0.002197} = 0.13\)
In simple words: Determine the number which when multiplied by itself three times gives the given value.

Exam Tip: Convert mixed fractions into improper fractions before taking cube roots.

Question 11. Evaluate
(i) \(\sqrt[3]{-729 \times (-216)}\)
(ii) \(\sqrt[3]{-500 \times -128}\)
(iii) \(\sqrt[3]{\frac{-9261}{15625}}\)
Answer:
(i) \(\sqrt[3]{-729 \times (-216)} = \sqrt[3]{729 \times 216} = \sqrt[3]{729} \times \sqrt[3]{216} = 9 \times 6 = 54\)
(ii) \(\sqrt[3]{-500 \times -128} = \sqrt[3]{500 \times 128} = \sqrt[3]{64000} = 40\)
(iii) \(\sqrt[3]{\frac{-9261}{15625}} = \frac{\sqrt[3]{-9261}}{\sqrt[3]{15625}} = \frac{-21}{25}\)
In simple words: Simplify the product under the cube root, then find the cube root of each part.

Exam Tip: Use the property \(\sqrt[3]{a \times b} = \sqrt[3]{a} \times \sqrt[3]{b}\) to simplify calculations.

Question 12. What is the edge of a cube whose volume is 46656 cm\(^3\)?
Answer:
Volume of cube = 46656 cm\(^3\)
Edge \((\text{a}) = \sqrt[3]{46656}\)
Prime factorization of \(46656 = 2^6 \times 3^6 = (2^2 \times 3^2)^3 = 36^3\)
Edge = 36 cm
In simple words: Take the cube root of the given volume to find the edge length.

Exam Tip: Volume = \(\text{edge}^3\), so Edge = \(\sqrt[3]{\text{Volume}}\).

Question 13. Find the cube root of 216 x 1331.
Answer:
\(\sqrt[3]{216 \times 1331} = \sqrt[3]{216} \times \sqrt[3]{1331} = 6 \times 11 = 66\)
In simple words: Find the cube roots of 216 and 1331 separately and multiply them.

Exam Tip: Splitting the cube root of a product makes calculation much faster.

Question 14. A rectangular cubical piece of metal of dimensions 2 cm x 3 cm x 4 cm is melted. Some more of the metal is added and it is made into a cube. The cube has integral measures for its sides. What is the minimum amount of metal that is added and what is the size of this cube.
Answer:
Volume of given cuboid = \(2 \times 3 \times 4 = 24\text{ cm}^3\)
The smallest perfect cube greater than 24 is \(27\) \((\text{since } 3^3 = 27)\).
Size of the new cube = 3 cm (since edge is an integer and volume is \(3^3 = 27\text{ cm}^3\))
Minimum metal added = \(27 - 24 = 3\text{ cm}^3\).
In simple words: Find the next perfect cube volume above 24, which is 27. The difference is the added metal.

Exam Tip: Volume remains constant during melting and recasting, except when extra metal is added.

Question 15. To collect rain water, Maria made a cubical tank which can hold 91125 m\(^3\) water. She uses this water for watering the plants of her garden.
(i) What is the height of the tank
(ii) By doing this, what is she promoting?
Answer:
(i) Volume of tank = \(91125\text{ m}^3\)
Since the tank is cubical, Height \(=\sqrt[3]{91125} = 45\text{ m}\).
(ii) She is promoting rain water harvesting and water conservation.
In simple words: The height of a cube is equal to the cube root of its volume. Saving rainwater helps conserve natural resources.

Exam Tip: For a cubical tank, length, breadth, and height are all equal.