Question
Given that the cube root of 27 is 3, what is 2743
?
To find the cube root of 2743, we can use the fact that the cube root of a product is equal to the product of the cube roots of the factors.
Prime factorizing 2743, we find that it is equal to 7 * 7 * 7 * 7 * 7.
Applying the cube root to each factor, we get the cube root of 7 * 7 * 7 * 7 * 7 is equal to the cube root of 7 * the cube root of 7 * the cube root of 7 * the cube root of 7 * the cube root of 7.
Since the cube root of 7 is not given, we cannot simplify it further. Therefore, the cube root of 2743 is written as 7 * 7 * 7 * 7 * 7 * the cube root of 7.
Given that the cube root of 27 is 3, what is 2743
?(1 point)
Responses
108
108
81
81
12
12
4
4
Using the fact that the cube root of a number multiplied by itself three times gives the original number, we can determine the cube root of 2743 by taking the cube root of the factors of 2743.
Prime factorizing 2743, we find that it is equal to 7 * 7 * 7 * 7 * 7.
Since the cube root of 7 is not given, we cannot further simplify it.
Therefore, the cube root of 2743 is 7 * 7 * 7 * 7 * 7 * the cube root of 7.
The correct answer is 7 * 7 * 7 * 7 * 7 * cube root of 7.
Which factors of the base would help simplify the expression 1614?(1 point)
Responses
To determine which factors of the base would help simplify the expression 1614, we need to find the prime factorization of 1614.
Prime factorizing 1614, we find that it is equal to 2 * 3 * 269.
When simplifying the expression, we should consider factors that can be simplified or canceled out.
In this case, there are no common factors that can be simplified or canceled out.
Therefore, the factors of the base 1614 do not help simplify the expression.
To find the cube root of 2743, you can first prime factorize the number.
Prime factorization of 2743:
2^3 × 7 × 31
Next, you can group the prime factors in sets of three to find the cube root.
(2^3) × (7) × (31)
The cube root of each group is:
(2^3) = 2
(7) = 7
(31) = 31
So the cube root of 2743 is 2 × 7 × 31 = 434.
To find the cube root of 2743, we can use a calculator or manually calculate it.
Method 1: Using a Calculator
Many calculators have a cube root function, often denoted as "∛" or "cube root". Simply enter 2743 and press the cube root button to get the result. In this case, the cube root of 2743 is approximately 14.843.
Method 2: Manual Calculation
To manually calculate the cube root of 2743, we can use estimation methods such as trial and error.
1. Start by guessing a number that might be close to the cube root of 2743. Since the cube root of 27 is 3, we can start with a number close to 27, such as 20.
2. Cube your guess: 20^3 = 8000.
3. Compare the result to 2743. If the result is too large, try a smaller number, and if it is too small, try a larger number.
4. To refine our guess, we can use a linear interpolation method. Divide the difference between the two numbers (in this case, 8000 - 2743 = 5257) by the difference between the cube of our initial guess and the cube of the refined guess. Let's say our refined guess is 21. We calculate: 20 + (21 - 20) * (2743 - 8000) / (2743 - 8000) = 20.121.
5. Cube the refined guess: 20.121^3 ≈ 8036.67.
6. As the result is still too large, we repeat the process by refining our guess again.
7. Keep repeating steps 4-6 until we obtain an accurate enough approximation.
By following this method repeatedly, we will eventually arrive at a value close to the actual cube root of 2743, which is approximately 14.843 (rounded to three decimal places).