Math worksheet with questions on cube roots and perfect cubes.

Problem Set Analysis and Solutions

The provided problem set involves cube calculations, cube roots, and operations with cubes. Below is a detailed solution for each question.

---

Question 1: Find the cube of

- a. \(-7\)
The cube of \(-7\) is calculated as:
\[
(-7)^3 = (-7) \times (-7) \times (-7) = -343
\]

- b. \(1 \frac{2}{3}\)
Convert the mixed fraction to an improper fraction:
\[
1 \frac{2}{3} = \frac{5}{3}
\]
The cube is:
\[
\left(\frac{5}{3}\right)^3 = \frac{5^3}{3^3} = \frac{125}{27}
\]

- c. \(2.5\)
The cube of \(2.5\) is:
\[
(2.5)^3 = 2.5 \times 2.5 \times 2.5 = 6.25 \times 2.5 = 15.625
\]

- d. \(-\frac{1}{15}\)
The cube of \(-\frac{1}{15}\) is:
\[
\left(-\frac{1}{15}\right)^3 = -\frac{1^3}{15^3} = -\frac{1}{3375}
\]

Final Answers for Question 1:
\[
\boxed{-343, \frac{125}{27}, 15.625, -\frac{1}{3375}}
\]

---

Question 2: Which of the following numbers are perfect cubes? In case of perfect cube, find the number whose cube is the given number.

- a. 243
Check if 243 is a perfect cube:
\[
\sqrt[3]{243} \approx 6.24 \quad (\text{not an integer})
\]
Therefore, 243 is not a perfect cube.

- b. 256
Check if 256 is a perfect cube:
\[
\sqrt[3]{256} \approx 6.35 \quad (\text{not an integer})
\]
Therefore, 256 is not a perfect cube.

- c. 5324
Check if 5324 is a perfect cube:
\[
\sqrt[3]{5324} \approx 17.46 \quad (\text{not an integer})
\]
Therefore, 5324 is not a perfect cube.

- d. 9261
Check if 9261 is a perfect cube:
\[
\sqrt[3]{9261} = 21 \quad (\text{an integer})
\]
Therefore, 9261 is a perfect cube, and the number whose cube is 9261 is \(21\).

Final Answer for Question 2:
\[
\boxed{9261 \text{ is a perfect cube, and } \sqrt[3]{9261} = 21}
\]

---

Question 3: Find the cube root of the following:

- a. \(-4096\)
\[
\sqrt[3]{-4096} = -16 \quad (\text{since } (-16)^3 = -4096)
\]

- b. \(-512\)
\[
\sqrt[3]{-512} = -8 \quad (\text{since } (-8)^3 = -512)
\]

- c. \(\frac{-3375}{1331}\)
\[
\sqrt[3]{\frac{-3375}{1331}} = \frac{\sqrt[3]{-3375}}{\sqrt[3]{1331}} = \frac{-15}{11} \quad (\text{since } (-15)^3 = -3375 \text{ and } 11^3 = 1331)
\]

- d. \(\frac{-512}{343}\)
\[
\sqrt[3]{\frac{-512}{343}} = \frac{\sqrt[3]{-512}}{\sqrt[3]{343}} = \frac{-8}{7} \quad (\text{since } (-8)^3 = -512 \text{ and } 7^3 = 343)
\]

- e. \(0.000729\)
\[
\sqrt[3]{0.000729} = 0.09 \quad (\text{since } (0.09)^3 = 0.000729)
\]

- f. \(1.728\)
\[
\sqrt[3]{1.728} = 1.2 \quad (\text{since } (1.2)^3 = 1.728)
\]

- g. \(0.017576\)
\[
\sqrt[3]{0.017576} = 0.26 \quad (\text{since } (0.26)^3 = 0.017576)
\]

- h. \(32.768\)
\[
\sqrt[3]{32.768} = 3.2 \quad (\text{since } (3.2)^3 = 32.768)
\]

- i. \(10648\)
\[
\sqrt[3]{10648} = 22 \quad (\text{since } 22^3 = 10648)
\]

- j. \(\frac{-3375}{2744}\)
\[
\sqrt[3]{\frac{-3375}{2744}} = \frac{\sqrt[3]{-3375}}{\sqrt[3]{2744}} = \frac{-15}{14} \quad (\text{since } (-15)^3 = -3375 \text{ and } 14^3 = 2744)
\]

Final Answers for Question 3:
\[
\boxed{-16, -8, \frac{-15}{11}, \frac{-8}{7}, 0.09, 1.2, 0.26, 3.2, 22, \frac{-15}{14}}
\]

---

Question 4: Find the smallest number by which the following numbers must be multiplied so that the product is a perfect cube. Also, find the cube root of the product.

- a. 1332
Prime factorize 1332:
\[
1332 = 2^2 \times 3^2 \times 37
\]
To make it a perfect cube, we need each prime factor's exponent to be a multiple of 3. We need one more \(2\), one more \(3\), and two more \(37\)'s:
\[
1332 \times 2 \times 3 \times 37^2 = 2^3 \times 3^3 \times 37^3 = (2 \times 3 \times 37)^3 = 222^3
\]
The smallest number to multiply is:
\[
2 \times 3 \times 37^2 = 2 \times 3 \times 1369 = 8214
\]
The cube root of the product is:
\[
\sqrt[3]{1332 \times 8214} = 222
\]

- b. 2560
Prime factorize 2560:
\[
2560 = 2^9 \times 5
\]
To make it a perfect cube, we need the exponent of 5 to be a multiple of 3. We need two more 5's:
\[
2560 \times 5^2 = 2^9 \times 5^3 = (2^3 \times 5)^3 = 40^3
\]
The smallest number to multiply is:
\[
5^2 = 25
\]
The cube root of the product is:
\[
\sqrt[3]{2560 \times 25} = 40
\]

- c. 35721
Prime factorize 35721:
\[
35721 = 3^4 \times 7^4
\]
To make it a perfect cube, we need each prime factor's exponent to be a multiple of 3. We need one more 3 and one more 7:
\[
35721 \times 3 \times 7 = 3^5 \times 7^5 = (3^2 \times 7^2)^3 = 441^3
\]
The smallest number to multiply is:
\[
3 \times 7 = 21
\]
The cube root of the product is:
\[
\sqrt[3]{35721 \times 21} = 441
\]

Final Answers for Question 4:
\[
\boxed{8214, 222; 25, 40; 21, 441}
\]

---

Question 5: What is the smallest number by which the following numbers must be divided so that the quotient is a perfect cube? Also, find the cube root of the quotient.

- a. 675
Prime factorize 675:
\[
675 = 3^3 \times 5^2
\]
To make it a perfect cube, we need the exponent of 5 to be a multiple of 3. We need to divide by \(5^2\):
\[
\frac{675}{5^2} = \frac{675}{25} = 27 = 3^3
\]
The smallest number to divide by is:
\[
25
\]
The cube root of the quotient is:
\[
\sqrt[3]{27} = 3
\]

- b. 8640
Prime factorize 8640:
\[
8640 = 2^6 \times 3^3 \times 5
\]
To make it a perfect cube, we need the exponent of 5 to be a multiple of 3. We need to divide by 5:
\[
\frac{8640}{5} = 1728 = 2^6 \times 3^3 = (2^2 \times 3)^3 = 12^3
\]
The smallest number to divide by is:
\[
5
\]
The cube root of the quotient is:
\[
\sqrt[3]{1728} = 12
\]

- c. 7803
Prime factorize 7803:
\[
7803 = 3^3 \times 17^2
\]
To make it a perfect cube, we need the exponent of 17 to be a multiple of 3. We need to divide by \(17^2\):
\[
\frac{7803}{17^2} = \frac{7803}{289} = 27 = 3^3
\]
The smallest number to divide by is:
\[
289
\]
The cube root of the quotient is:
\[
\sqrt[3]{27} = 3
\]

Final Answers for Question 5:
\[
\boxed{25, 3; 5, 12; 289, 3}
\]

---

Question 6: Evaluate:

- a. \(\sqrt[3]{36} \times \sqrt[3]{384}\)
Use the property \(\sqrt[3]{a} \times \sqrt[3]{b} = \sqrt[3]{a \times b}\):
\[
\sqrt[3]{36} \times \sqrt[3]{384} = \sqrt[3]{36 \times 384} = \sqrt[3]{13824}
\]
Prime factorize 13824:
\[
13824 = 2^9 \times 3^3 = (2^3 \times 3)^3 = 24^3
\]
Therefore:
\[
\sqrt[3]{13824} = 24
\]

- b. \(\sqrt[3]{96} \times \sqrt[3]{144}\)
Use the property \(\sqrt[3]{a} \times \sqrt[3]{b} = \sqrt[3]{a \times b}\):
\[
\sqrt[3]{96} \times \sqrt[3]{144} = \sqrt[3]{96 \times 144} = \sqrt[3]{13824}
\]
From part (a), we know:
\[
\sqrt[3]{13824} = 24
\]

- c. \(\sqrt[3]{100} \times \sqrt[3]{270}\)
Use the property \(\sqrt[3]{a} \times \sqrt[3]{b} = \sqrt[3]{a \times b}\):
\[
\sqrt[3]{100} \times \sqrt[3]{270} = \sqrt[3]{100 \times 270} = \sqrt[3]{27000}
\]
Prime factorize 27000:
\[
27000 = 2^3 \times 3^3 \times 5^3 = (2 \times 3 \times 5)^3 = 30^3
\]
Therefore:
\[
\sqrt[3]{27000} = 30
\]

- d. \(\sqrt[3]{121} \times \sqrt[3]{289}\)
Use the property \(\sqrt[3]{a} \times \sqrt[3]{b} = \sqrt[3]{a \times b}\):
\[
\sqrt[3]{121} \times \sqrt[3]{289} = \sqrt[3]{121 \times 289}
\]
Note that 121 and 289 are not perfect cubes, and their product is also not a perfect cube. Therefore, the expression remains:
\[
\sqrt[3]{121 \times 289}
\]

- e. \(\sqrt[3]{-27} \times \sqrt[3]{2744}\)
Use the property \(\sqrt[3]{a} \times \sqrt[3]{b} = \sqrt[3]{a \times b}\):
\[
\sqrt[3]{-27} \times \sqrt[3]{2744} = \sqrt[3]{-27 \times 2744}
\]
Calculate the product:
\[
-27 \times 2744 = -74088
\]
Prime factorize 74088:
\[
74088 = 2^3 \times 3^3 \times 7^3 = (2 \times 3 \times 7)^3 = 42^3
\]
Therefore:
\[
\sqrt[3]{-74088} = -42
\]

- f. \(\sqrt[3]{729} \times \sqrt[3]{-15625}\)
Use the property \(\sqrt[3]{a} \times \sqrt[3]{b} = \sqrt[3]{a \times b}\):
\[
\sqrt[3]{729} \times \sqrt[3]{-15625} = \sqrt[3]{729 \times -15625}
\]
Calculate the product:
\[
729 \times -15625 = -11390625
\]
Prime factorize 11390625:
\[
11390625 = 3^6 \times 5^6 = (3^2 \times 5^2)^3 = 225^3
\]
Therefore:
\[
\sqrt[3]{-11390625} = -225
\]

- g. \(\sqrt[3]{\frac{-125}{343}}\)
Use the property \(\sqrt[3]{\frac{a}{b}} = \frac{\sqrt[3]{a}}{\sqrt[3]{b}}\):
\[
\sqrt[3]{\frac{-125}{343}} = \frac{\sqrt[3]{-125}}{\sqrt[3]{343}} = \frac{-5}{7}
\]

- h. \(\sqrt[3]{512} \times \sqrt[3]{729}\)
Use the property \(\sqrt[3]{a} \times \sqrt[3]{b} = \sqrt[3]{a \times b}\):
\[
\sqrt[3]{512} \times \sqrt[3]{729} = \sqrt[3]{512 \times 729}
\]
Calculate the product:
\[
512 \times 729 = 373248
\]
Prime factorize 373248:
\[
373248 = 2^9 \times 3^6 = (2^3 \times 3^2)^3 = 72^3
\]
Therefore:
\[
\sqrt[3]{373248} = 72
\]

- i. \(\sqrt[3]{(-1331)} \times 3375\)
First, find \(\sqrt[3]{-1331}\):
\[
\sqrt[3]{-1331} = -11 \quad (\text{since } (-11)^3 = -1331)
\]
Then multiply by 3375:
\[
-11 \times 3375 = -37125
\]

- j. \(\sqrt[3]{744} \times (-4913)\)
First, find \(\sqrt[3]{744}\):
\[
\sqrt[3]{744} \approx 9.06 \quad (\text{not an integer})
\]
Then multiply by \(-4913\):
\[
\sqrt[3]{744} \times (-4913) \approx 9.06 \times (-4913) = -44588.78
\]

Final Answers for Question 6:
\[
\boxed{24, 24, 30, \sqrt[3]{121 \times 289}, -42, -225, \frac{-5}{7}, 72, -37125, -44588.78}
\]

---

Final Boxed Answers:

1. \(\boxed{-343, \frac{125}{27}, 15.625, -\frac{1}{3375}}\)
2. \(\boxed{9261 \text{ is a perfect cube, and } \sqrt[3]{9261} = 21}\)
3. \(\boxed{-16, -8, \frac{-15}{11}, \frac{-8}{7}, 0.09, 1.2, 0.26, 3.2, 22, \frac{-15}{14}}\)
4. \(\boxed{8214, 222; 25, 40; 21, 441}\)
5. \(\boxed{25, 3; 5, 12; 289, 3}\)
6. \(\boxed{24, 24, 30, \sqrt[3]{121 \times 289}, -42, -225, \frac{-5}{7}, 72, -37125, -44588.78}\)