The value of \(\frac{\sqrt[3]{-2744} \times \sqrt[3]{-216}}{\sqrt[3]{\frac{64}{729}}}\) is:
Solving Expressions with Cube Roots
The question asks us to find the value of a mathematical expression involving cube roots. The expression is:
\(\frac{\sqrt[3]{-2744} \times \sqrt[3]{-216}}{\sqrt[3]{\frac{64}{729}}}\)
To solve this, we need to evaluate each cube root separately and then perform the indicated multiplication and division.
Step-by-Step Calculation of Cube Roots
Let's calculate each part of the expression:
- Calculate the first cube root in the numerator: \(\sqrt[3]{-2744}\)
The cube root of a negative number is negative. We need to find a number \(x\) such that \(x^3 = -2744\). This means we need to find a number \(y\) such that \(y^3 = 2744\), and then the result will be \(-y\).
We can test small integer cubes or use prime factorization.
Let's find the cube root of 2744.
\(10^3 = 1000\)
\(20^3 = 8000\)
The number is between 10 and 20. The last digit of 2744 is 4. A number whose cube ends in 4 must end in 4 (since \(4^3 = 64\)). Let's try 14.
\(14^3 = 14 \times 14 \times 14 = 196 \times 14\)
\(196 \times 14 = 196 \times (10 + 4) = 196 \times 10 + 196 \times 4 = 1960 + 784 = 2744\)
So, \(\sqrt[3]{2744} = 14\).
Therefore, \(\sqrt[3]{-2744} = -14\). - Calculate the second cube root in the numerator: \(\sqrt[3]{-216}\)
Again, the cube root of a negative number is negative. We need a number \(x\) such that \(x^3 = -216\). Find \(y\) such that \(y^3 = 216\).
\(5^3 = 125\)
\(6^3 = 216\)
So, \(\sqrt[3]{216} = 6\).
Therefore, \(\sqrt[3]{-216} = -6\). - Calculate the cube root in the denominator: \(\sqrt[3]{\frac{64}{729}}\)
The cube root of a fraction is the cube root of the numerator divided by the cube root of the denominator: \(\sqrt[3]{\frac{a}{b}} = \frac{\sqrt[3]{a}}{\sqrt[3]{b}}\).
So, \(\sqrt[3]{\frac{64}{729}} = \frac{\sqrt[3]{64}}{\sqrt[3]{729}}\).
Calculate the numerator: \(\sqrt[3]{64}\). We know \(4^3 = 64\), so \(\sqrt[3]{64} = 4\).
Calculate the denominator: \(\sqrt[3]{729}\). We know \(9^3 = 729\), so \(\sqrt[3]{729} = 9\).
Therefore, \(\sqrt[3]{\frac{64}{729}} = \frac{4}{9}\).
Combining the Results
Now substitute the calculated values back into the original expression:
\(\frac{\sqrt[3]{-2744} \times \sqrt[3]{-216}}{\sqrt[3]{\frac{64}{729}}} = \frac{(-14) \times (-6)}{\frac{4}{9}}\)
First, calculate the numerator:
\((-14) \times (-6)\)
Multiplying two negative numbers gives a positive result.
\(14 \times 6 = 84\)
So, \((-14) \times (-6) = 84\).
Now, perform the division:
\(\frac{84}{\frac{4}{9}}\)
Dividing by a fraction is the same as multiplying by its reciprocal.
\(\frac{84}{\frac{4}{9}} = 84 \times \frac{9}{4}\)
We can simplify by dividing 84 by 4:
\(84 \div 4 = 21\)
So the expression becomes:
\(21 \times 9\)
\(21 \times 9 = 189\)
Thus, the value of the expression is 189.
Final Answer Verification
Let's quickly review the steps and calculations:
- \(\sqrt[3]{-2744} = -14\)
- \(\sqrt[3]{-216} = -6\)
- \(\sqrt[3]{\frac{64}{729}} = \frac{4}{9}\)
- Numerator product: \((-14) \times (-6) = 84\)
- Expression value: \(\frac{84}{\frac{4}{9}} = 84 \times \frac{9}{4} = 21 \times 9 = 189\)
The calculated value is 189, which corresponds to one of the given options.
| Part of Expression | Calculation | Result |
|---|---|---|
| \(\sqrt[3]{-2744}\) | Cube root of -2744 | -14 |
| \(\sqrt[3]{-216}\) | Cube root of -216 | -6 |
| \(\sqrt[3]{\frac{64}{729}}\) | Cube root of 64/729 | 4/9 |
| Numerator | \((-14) \times (-6)\) | 84 |
| Final Value | \(\frac{84}{4/9} = 84 \times \frac{9}{4}\) | 189 |
Revision Table: Understanding Cube Roots
| Concept | Description | Example |
|---|---|---|
| Cube Root | The number that, when multiplied by itself three times, gives the original number. Symbol: \(\sqrt[3]{\phantom{x}}\) | \(\sqrt[3]{8} = 2\) because \(2 \times 2 \times 2 = 8\) |
| Cube Root of Negative Number | The cube root of a negative number is always a real negative number. | \(\sqrt[3]{-8} = -2\) because \((-2) \times (-2) \times (-2) = -8\) |
| Cube Root of Fraction | \(\sqrt[3]{\frac{a}{b}} = \frac{\sqrt[3]{a}}{\sqrt[3]{b}}\) (where \(b \neq 0\)) | \(\sqrt[3]{\frac{1}{27}} = \frac{\sqrt[3]{1}}{\sqrt[3]{27}} = \frac{1}{3}\) |
| Multiplication of Signed Numbers | Negative \(\times\) Negative = Positive Negative \(\times\) Positive = Negative |
\((-5) \times (-3) = 15\) \((-5) \times 3 = -15\) |
| Dividing by a Fraction | Multiplying by the reciprocal of the divisor. \(\frac{a}{b/c} = a \times \frac{c}{b}\) | \(\frac{10}{1/2} = 10 \times 2 = 20\) |
Additional Information on Cube Root Calculations
Finding the cube root of larger numbers often involves recognizing perfect cubes or using prime factorization. For instance, to find \(\sqrt[3]{2744}\):
Prime factorization of 2744:
- \(2744 \div 2 = 1372\)
- \(1372 \div 2 = 686\)
- \(686 \div 2 = 343\)
- \(343 \div 7 = 49\)
- \(49 \div 7 = 7\)
- \(7 \div 7 = 1\)
So, \(2744 = 2 \times 2 \times 2 \times 7 \times 7 \times 7 = 2^3 \times 7^3\).
\(\sqrt[3]{2744} = \sqrt[3]{2^3 \times 7^3} = \sqrt[3]{(2 \times 7)^3} = 2 \times 7 = 14\).
Similarly, for 729:
Prime factorization of 729:
- \(729 \div 3 = 243\)
- \(243 \div 3 = 81\)
- \(81 \div 3 = 27\)
- \(27 \div 3 = 9\)
- \(9 \div 3 = 3\)
- \(3 \div 3 = 1\)
So, \(729 = 3 \times 3 \times 3 \times 3 \times 3 \times 3 = 3^6 = (3^2)^3 = 9^3\).
\(\sqrt[3]{729} = \sqrt[3]{9^3} = 9\).
This confirms the cube root calculations used in the solution.
Important Questions from Cube and Cube Root
If 5 \(\sqrt{3}\) + \(\sqrt{75}\) = 17.32, then the value of 14 \(\sqrt{3}\) + \(\sqrt{108}\) is:
Find the value of (25)3 + (-29)3 + (4)3
A number is cube of 53. When 7 times of 57 is subtracted from the number, then the resultant number which is formed will be divisible by:
The sum of a positive number and its cube is 1740. What is the value of the number?
The largest four digit number which is a perfect cube is:
