CBSE Class 8 Mathematics Cubes And Cube Roots Worksheet Set 02

Question. Divide the number 26244 by the smallest number so that the question is a perfect cube, so the smallest number is :
(A) 4
(B) 6
(C) 36
(D) 16

Answer: C

Question. 55 x 55 – 2 x 25 x 55 + 25 x 25/302 = ?
(A) 6
(B) 4
(C) 5
(D) None of these

Answer: D

Question. The volumes of two cubes are in the ratio of 343 : 1331, the ratio of their edges is .
(A) 7: 10
(B) 7:11
(C) 7: 12
(D) None of these

Answer: B

Question. The least number which must be added to 4931 to make it perfect square is :
(A) 110
(B) 120
(C) 130
(D) None

Answer: A

Question. The least number which must be subtracted from 18265 to make it a perfect square is:
(A) 20
(B) 30
(C) 40
(D) 50

Answer: C

Question. √2√2√2√2√2 is equal to
(A) 0
(B) 2
(C) 1
(D) 2 31/32

Answer: D

Question. 65 x 65+ 2 x 65 x 35 + 35 x 35/65 + 35 = ?
(A) 1000
(B) 10
(C) 100
(D) None of these

Answer: C

Question. Value of 3√392 x 3√448 is
(A) 50
(B) 52
(C) 54
(D) 56

Answer: D

Question. The value of √225 is-
(A) 25
(B) 15
(C) 35
(D) 05

Answer: B

Question. The smallest number by which 137592 should be multiply to make it perfect cube.
(A) 1183
(B) 1180
(C) 1181
(D) None of these

Answer: A

Question. (48 - 8)2/40 = ?
(A) 40
(B) 80
(C) 15
(D) 150

Answer: A

Question. By studying the numbers given below, find out the missing numbers.
(1) 2+ (2)² + (2)² = (3)²
(i) (4)² + (5)² + __² = (21)²
(A) 20 (B) 6 (C) 7 (D) 24
(ii) (5)² + __² + (30)² = (31)²

Answer: A

Question. 'The greatest six digit number which is perfect square is :
(A) 998004
(B) 998006
(C) 998049
(D) 998001

Answer: D

Question. A gardner plants 17956 trees in such a way that there are as many rows as there are trees in each row. The number of trees in a row are :
(A) 136
(B) 164
(C) 134
(D) 166

Answer: C

Question. (65 - 56)²/3² = ?
(A) 8
(B) 9
(C) 1/9
(D) None of these

Answer: B

Question. (16)² x (4)² = ?
(A) 4096
(B) 16360
(C) 25360
(D) None of these

Answer: A

Question. (15/8 x 16)² = ?
(A) 916
(B) 900
(C) 926
(D) 1966

Answer: B

Question. (16)² + (12)²/(20)² = ?
(A) 4
(B) 2
(C) 1
(D) 5

Answer: C

Question. Least perfect square of 6 digits is :
(A) 998000
(B) 998001
(C) 998002
(D) None of these

Answer: B

Question. (65)² – (64)² + (1)² =?
(A) 130
(B) 129
(C) 138
(D) 120

Answer: A

Question. (57)² – (52)² = 5 × ?
(A) 109
(B) 108
(C) 5
(D) None of these

Answer: A

Question. (5, 12, 15) and (7, 5, 12)
(A) Both are Pythagorean triplets
(B) None of the both is Pythagorean triplet
(C) First is Pythagorean triplet while second is not
(D) Second is Pythagorean triplet while first is not

Answer: B

Question. The perfect cube nearest to 2750 is :
(A) 2749
(B) 2747
(C) 2744
(D) 2754

Answer: C

Question. If (65)² – (60)² = [(55)² – (50)² ] × y, then the value of y is-
(A) 23/21
(B) 28/21
(C) 25/21
(D) None of these

Answer: C

Question. A 8 × 6 × 4 cm³ metallic cube is melted. Find the minimum volume of molten metal which should be added to mould it into a cube whose edge is an x, where x is an integer.
(A) 20
(B) 21
(C) 23
(D) 24

Answer: D

Chapter: 7 Cubes and Cube Roots

Question 1. Write the unit's digits of the cube of each of the following numbers:
(i) 31      (ii) 109      (iii) 388      (iv) 22      (v) 30
(vi) 83      (vii) 24      (viii) 65      (ix) 76      (x) 107
Answer: To find the unit's digit of the cube of a number, we only need to cube the unit's digit of the given number and look at the unit's digit of that result.
(i) For 31, the unit's digit is 1. Since \(1^3 = 1\), the unit's digit is 1.
(ii) For 109, the unit's digit is 9. Since \(9^3 = 729\), the unit's digit is 9.
(iii) For 388, the unit's digit is 8. Since \(8^3 = 512\), the unit's digit is 2.
(iv) For 22, the unit's digit is 2. Since \(2^3 = 8\), the unit's digit is 8.
(v) For 30, the unit's digit is 0. Since \(0^3 = 0\), the unit's digit is 0.
(vi) For 83, the unit's digit is 3. Since \(3^3 = 27\), the unit's digit is 7.
(vii) For 24, the unit's digit is 4. Since \(4^3 = 64\), the unit's digit is 4.
(viii) For 65, the unit's digit is 5. Since \(5^3 = 125\), the unit's digit is 5.
(ix) For 76, the unit's digit is 6. Since \(6^3 = 216\), the unit's digit is 6.
(x) For 107, the unit's digit is 7. Since \(7^3 = 343\), the unit's digit is 3.
In simple words: Only the last digit of a number determines the last digit of its cube.

Exam Tip: Memorize the cubes of digits from 0 to 9 to quickly find unit digits without full multiplication.

Question 2. Find the cube of the following numbers:
(i) 12      (ii) \(\frac{9}{10}\)      (iii) -31      (iv) 0.5      (v) \(\frac{2}{3}\)
Answer:
(i) Cube of 12 = \(12^3 = 12 \times 12 \times 12 = 144 \times 12 = 1728\)
(ii) Cube of \(\frac{9}{10}\) = \(\left(\frac{9}{10}\right)^3 = \frac{9^3}{10^3} = \frac{729}{1000} = 0.729\)
(iii) Cube of -31 = \((-31)^3 = (-31) \times (-31) \times (-31) = 961 \times (-31) = -29791\)
(iv) Cube of 0.5 = \((0.5)^3 = 0.5 \times 0.5 \times 0.5 = 0.125\)
(v) Cube of \(\frac{2}{3}\) = \(\left(\frac{2}{3}\right)^3 = \frac{2^3}{3^3} = \frac{8}{27}\)
In simple words: Multiply the number by itself three times, keeping track of negative signs.

Exam Tip: The cube of a negative number is always negative, and the cube of a fraction is found by cubing both numerator and denominator separately.

Question 3. Check whether the following numbers are perfect cubes:
(i) 3375      (ii) 5832      (iii) 8000      (iv) 14400
Answer: To check if a number is a perfect cube, we find its prime factorization and check if all prime factors can be grouped in triplets.
(i) For 3375:
\(3375 = 3 \times 3 \times 3 \times 5 \times 5 \times 5 = 3^3 \times 5^3 = (3 \times 5)^3 = 15^3\)
Since all prime factors form complete triplets, 3375 is a perfect cube.
(ii) For 5832:
\(5832 = 2 \times 2 \times 2 \times 3 \times 3 \times 3 \times 3 \times 3 \times 3 = 2^3 \times 3^3 \times 3^3 = (2 \times 3 \times 3)^3 = 18^3\)
Since all prime factors form complete triplets, 5832 is a perfect cube.
(iii) For 8000:
\(8000 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 5 \times 5 \times 5 = 2^3 \times 2^3 \times 5^3 = (2 \times 2 \times 5)^3 = 20^3\)
Since all prime factors form complete triplets, 8000 is a perfect cube.
(iv) For 14400:
\(14400 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 3 \times 3 \times 5 \times 5 = 2^3 \times 2^3 \times 3^2 \times 5^2\)
Here, 3 and 5 do not form complete triplets. Therefore, 14400 is not a perfect cube.
In simple words: A number is a perfect cube if every prime factor in its breakdown appears in groups of three.

Exam Tip: Always use prime factorization and group identical factors in sets of three to test for perfect cubes.

Question 4. Find the least number by which the following numbers multiplied to make them perfect cubes:
(i) 392      (ii) 675      (iii) 1024      (iv) 5400      (v) 3136      (vi) 1323
Answer:
(i) For 392:
\(392 = 2 \times 2 \times 2 \times 7 \times 7 = 2^3 \times 7^2\)
The prime factor 7 has a power of 2, so it needs one more 7 to form a triplet. Therefore, the least number to multiply is 7.
(ii) For 675:
\(675 = 3 \times 3 \times 3 \times 5 \times 5 = 3^3 \times 5^2\)
The prime factor 5 needs one more 5 to form a triplet. Therefore, the least number to multiply is 5.
(iii) For 1024:
\(1024 = 2^{10} = 2^3 \times 2^3 \times 2^3 \times 2\)
The prime factor 2 needs two more 2s (i.e., \(2^2 = 4\)) to form a triplet. Therefore, the least number to multiply is 4.
(iv) For 5400:
\(5400 = 2^3 \times 3^3 \times 5^2\)
The prime factor 5 needs one more 5 to form a triplet. Therefore, the least number to multiply is 5.
(v) For 3136:
\(3136 = 2^6 \times 7^2 = 2^3 \times 2^3 \times 7^2\)
The prime factor 7 needs one more 7 to form a triplet. Therefore, the least number to multiply is 7.
(vi) For 1323:
\(1323 = 3^3 \times 7^2\)
The prime factor 7 needs one more 7 to form a triplet. Therefore, the least number to multiply is 7.
In simple words: Find the prime factors, check which ones are short of a group of three, and multiply by that missing amount.

Exam Tip: Identify deficient prime factors whose exponents are not multiples of 3.

Question 5. By what numbers (least) the above numbers should be divided to make them perfect cubes?
Answer: We divide by the leftover prime factors that do not form complete triplets.
(i) For 392 (\(2^3 \times 7^2\)): Divide by \(7^2 = 49\).
(ii) For 675 (\(3^3 \times 5^2\)): Divide by \(5^2 = 25\).
(iii) For 1024 (\(2^9 \times 2\)): Divide by 2.
(iv) For 5400 (\(2^3 \times 3^3 \times 5^2\)): Divide by \(5^2 = 25\).
(v) For 3136 (\(2^6 \times 7^2\)): Divide by \(7^2 = 49\).
(vi) For 1323 (\(3^3 \times 7^2\)): Divide by \(7^2 = 49\).
In simple words: Divide the number by whatever extra factors are left over outside of complete triplets.

Exam Tip: The number to divide by is always the product of the incomplete prime factor groups.

Question 6. Find the cube roots of the following:
(i) 15625      (ii) 35937      (iii) 74088
(iv) -343 / 729      (v) \(18\frac{26}{27}\)
Answer:
(i) For 15625:
\(15625 = 5 \times 5 \times 5 \times 5 \times 5 \times 5 = 5^3 \times 5^3 = (25)^3\)
\(\sqrt[3]{15625} = 25\)
(ii) For 35937:
\(35937 = 3 \times 3 \times 3 \times 11 \times 11 \times 11 = 3^3 \times 11^3 = (33)^3\)
\(\sqrt[3]{35937} = 33\)
(iii) For 74088:
\(74088 = 2^3 \times 3^3 \times 7^3 = (2 \times 3 \times 7)^3 = 42^3\)
\(\sqrt[3]{74088} = 42\)
(iv) For \(-\frac{343}{729}\):
\(\sqrt[3]{-\frac{343}{729}} = \frac{\sqrt[3]{-343}}{\sqrt[3]{729}} = \frac{-7}{9}\)
(v) For \(18\frac{26}{27}\):
First convert the mixed fraction to an improper fraction:
\(18\frac{26}{27} = \frac{18 \times 27 + 26}{27} = \frac{486 + 26}{27} = \frac{512}{27}\)
\(\sqrt[3]{\frac{512}{27}} = \frac{\sqrt[3]{512}}{\sqrt[3]{27}} = \frac{8}{3}\)
In simple words: Break each number into prime factors, group them into triplets, and take one factor from each group.

Exam Tip: For mixed fractions, always convert to improper fractions before prime factorizing or taking cube roots.

Question 7. Evaluate: \(\sqrt[3]{216} \times \sqrt[3]{729}\)
Answer:
\(\sqrt[3]{216} = \sqrt[3]{6^3} = 6\)
\(\sqrt[3]{729} = \sqrt[3]{9^3} = 9\)
\(\sqrt[3]{216} \times \sqrt[3]{729} = 6 \times 9 = 54\)
In simple words: Find the cube root of each number separately and then multiply the results.

Exam Tip: You can either multiply numbers first under a single cube root or evaluate them separately if they are standard cubes.

Question 8. What is the edge of a cube whose volume is \(13824 \text{ cm}^3\)?
Answer:
Volume of cube = \((\text{edge})^3\)
\((\text{edge})^3 = 13824\)
Edge = \(\sqrt[3]{13824}\)
By prime factorization: \(13824 = 2^9 \times 3^3 = (2^3 \times 2^3 \times 3)^3 = (24)^3\)
Edge = 24 cm
In simple words: To find the side length of a cube from its volume, calculate the cube root of the volume.

Exam Tip: Do not forget to include the correct units (cm) in the final answer for edge length.

Question 9. Find the volume of a cube whose edge is \(2.5 \text{ cm}\).
Answer:
Volume of cube = \((\text{edge})^3\)
Volume = \((2.5)^3 = 2.5 \times 2.5 \times 2.5 = 15.625 \text{ cm}^3\)
In simple words: Cube the given edge length to get the volume.

Exam Tip: Volume is always expressed in cubic units (\(\text{cm}^3\)).

Question 10. Find the cube roots of the following numbers by estimation:
(i) 15625      (ii) 35937      (iii) 42875
Answer:
(i) For 15625:
Step 1: Form two groups: 15 and 625.
Step 2: The unit digit is 5, so the unit digit of the cube root is 5.
Step 3: For the first group 15, since \(2^3 = 8\) and \(3^3 = 27\), the tens digit is 2.
Therefore, \(\sqrt[3]{15625} = 25\).

(ii) For 35937:
Step 1: Form two groups: 35 and 937.
Step 2: The unit digit is 7, so the unit digit of the cube root is 3 (since \(3^3 = 27\)).
Step 3: For the first group 35, since \(3^3 = 27\) and \(4^3 = 64\), the tens digit is 3.
Therefore, \(\sqrt[3]{35937} = 33\).

(iii) For 42875:
Step 1: Form two groups: 42 and 875.
Step 2: The unit digit is 5, so the unit digit of the cube root is 5.
Step 3: For the first group 42, since \(3^3 = 27\) and \(4^3 = 64\), the tens digit is 3.
Therefore, \(\sqrt[3]{42875} = 35\).
In simple words: Split the number into two parts, estimate the unit digit from the last digit, and estimate the tens digit from the first group.

Exam Tip: The estimation method is fast and relies heavily on knowing the unit digits of cubes from 1 to 9.