Answer :
Answer:
Let's address each question step by step:
**7. Evaluate.**
(a) \((5/11)^3 (8/15)^3 (13/21)^3\)
To evaluate this expression, we can combine all the fractions and then cube the result:
\[
\left(\frac{5 \cdot 8 \cdot 13}{11 \cdot 15 \cdot 21}\right)^3
\]
Calculate the numerator and the denominator separately:
Numerator: \(5 \cdot 8 \cdot 13 = 520\)
Denominator: \(11 \cdot 15 \cdot 21 = 3465\)
So, the expression simplifies to:
\[
\left(\frac{520}{3465}\right)^3
\]
(b) \((0.8/21)^3\)
\[
\left(\frac{0.8}{21}\right)^3 = \left(\frac{8}{210}\right)^3 = \left(\frac{4}{105}\right)^3
\]
(c) \((23/27)^3\)
\[
\left(\frac{23}{27}\right)^3
\]
**8. Using prime factorisation, check if the following numbers are perfect cubes.**
(a) \(15,625 = 5^6\) (which is \(5^3\)^2, so it is a perfect cube).
(b) Prime factorization of \(10,648\) and others needs to be checked.
**9. Find the cube root of:**
(a) \(2,87,496\)
(b) \(2,19,52,000\)
(c) \(74,088\)
(d) \(3,73,248\)
**10. Find the cube root of:**
(a) \(18 \frac{26}{27}\)
(b) \(-0.000216\)
(c) \(2 \frac{82}{1331}\)
(d) \(13.824\)
**11. If \(x = \sqrt{0.000064}\), find \( \sqrt[3]{x} \).**
**12. Find the value of \( \sqrt{968 \times \sqrt{1375 \sqrt[3]{1331}}} \).**
**13. Write cubes of five natural numbers which are multiples of 3 and verify the following: The cube of a natural number which is a multiple of 3 is a multiple of 27.**
**14. Prime factorisation of a number is \(2^5 \times 3^6 \times 7^2\). Is this number a perfect cube? Find the least number by which it should be multiplied so that the resulting number is a perfect cube.**
**15. Is \(13,720\) a perfect cube? If no, find the smallest number by which it should be multiplied so that the resulting number is a perfect cube.**
**16. Find the smallest multiple of 3, 4, and 6 which is a perfect cube.**
**17. Is the largest 3-digit number a perfect cube? If no, find the least number by which it should be multiplied so that it becomes a perfect cube.**
**18. By what least number should the smallest 5-digit number be divided so that the resulting number is a perfect cube?**
**19. Divide \(43,200\) by the smallest number so that the quotient is a perfect cube. What is that number? Also find the cube root of the quotient.**
To fully solve these questions, each one requires detailed calculations, factorizations, and sometimes, algorithms or numerical methods for precise answers. Let me know if you want to focus on specific questions or need detailed solutions for any particular one!
Answer:
Here are the solutions to all the questions:
1. Evaluate:
(a) (5/11)³ (8/15)³ (13/21)³ = (125/1331) (512/3375) (2197/9261) = 1/243
(d) (0.8/21)³ = (8/210)³ = 512/926100
(e) (23/27)³ = 12167/19683
1. Using prime factorization:
(a) 15,625 = 5^4 × 5^4 (not a perfect cube)
(b) 10,648 = 2^3 × 2^3 × 11^3 (perfect cube)
(c) 53,240 = 2^3 × 2^3 × 5 × 11^3 (not a perfect cube)
(d) 74,088 = 2^3 × 2^3 × 3^3 × 17^3 (perfect cube)
1. Find the cube root:
(a) ∛2,87,496 = 66
(b) ∛2,19,52,000 = 60
(c) ∛74,088 = 42
(d) ∛3,73,248 = 156
1. Find the cube root:
(a) ∛18 26/27 = ∛(18 26/27) = 2 6/9
(b) ∛-0.000216 = -0.06
(c) ∛2 82/1331 = ∛(2 82/1331) = 1 4/11
(d) ∛13.824 = 2.4
1. If x = sqrt(0.000064), then root(x, 3) = ∛(0.000064) = 0.04
2. sqrt[968 × sqrt[1375 × root(1331, 3)] = sqrt(968 × sqrt(1375 × 11)) = sqrt(968 × 37) = 56
3. Cubes of five natural numbers which are multiples of 3:
- 3^3 = 27
- 6^3 = 216
- 9^3 = 729
- 12^3 = 1728
- 15^3 = 3375
All of these cubes are multiples of 27, verifying the statement.
1. The given number is not a perfect cube. To make it a perfect cube, it should be multiplied by 2^2 × 3^3 × 7^2 = 1323.
2. 13,720 is not a perfect cube. To make it a perfect cube, it should be multiplied by 10.
3. The smallest multiple of 3, 4, and 6 which is a perfect cube is 216.
4. The largest 3-digit number (999) is not a perfect cube. To make it a perfect cube, it should be multiplied by 4.
5. The smallest 5-digit number (10,000) should be divided by 4 to get a perfect cube.
6. Dividing 43,200 by 27 gives a perfect cube. The quotient is 1600, and its cube root is 12.
Step-by-step explanation:
Here is a step-by-step explanation for each question:
1. Evaluate:
(a) (5/11)³ (8/15)³ (13/21)³
= (125/1331) × (512/3375) × (2197/9261)
= 1/243
(d) (0.8/21)³
= (8/210)³
= 512/926100
(e) (23/27)³
= 12167/19683
1. Using prime factorization:
(a) 15,625
= 5^4 × 5^4 (not a perfect cube)
(b) 10,648
= 2^3 × 2^3 × 11^3 (perfect cube)
(c) 53,240
= 2^3 × 2^3 × 5 × 11^3 (not a perfect cube)
(d) 74,088
= 2^3 × 2^3 × 3^3 × 17^3 (perfect cube)
1. Find the cube root:
(a) ∛2,87,496
= 66
(b) ∛2,19,52,000
= 60
(c) ∛74,088
= 42
(d) ∛3,73,248
= 156
1. Find the cube root:
(a) ∛18 26/27
= ∛(18 26/27)
= 2 6/9
(b) ∛-0.000216
= -0.06
(c) ∛2 82/1331
= ∛(2 82/1331)
= 1 4/11
(d) ∛13.824
= 2.4
1. If x = sqrt(0.000064), then root(x, 3)
= ∛(0.000064)
= 0.04
2. sqrt[968 × sqrt[1375 × root(1331, 3)]
= sqrt(968 × sqrt(1375 × 11))
= sqrt(968 × 37)
= 56
3. Cubes of five natural numbers which are multiples of 3:
- 3^3 = 27
- 6^3 = 216
- 9^3 = 729
- 12^3 = 1728
- 15^3 = 3375
All of these cubes are multiples of 27, verifying the statement.
1. The given number is not a perfect cube. To make it a perfect cube, it should be multiplied by 2^2 × 3^3 × 7^2 = 1323.
2. 13,720 is not a perfect cube. To make it a perfect cube, it should be multiplied by 10.
3. The smallest multiple of 3, 4, and 6 which is a perfect cube is 216.
4. The largest 3-digit number (999) is not a perfect cube. To make it a perfect cube, it should be multiplied by 4.
5. The smallest 5-digit number (10,000) should be divided by 4 to get a perfect cube.
6. Dividing 43,200 by 27 gives a perfect cube. The quotient is 1600, and its cube root is 12.
Let me know if you need further clarification on any of these steps!
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