Find the cubic root of 56: ∛56
To find the cubic root of 56, you can follow these steps:
Step 1: Find a number that, when cubed (raised to the power of 3), gives you a value closest to 56. Begin with an estimation by using the integers. ∛56 is somewhere between ∛27 (3^3) and ∛64 (4^3). The closest value is 3.
Step 2: To refine the estimation, divide 56 by the estimated value from Step 1 (3).
\( \frac{56}{3} = 18.6667 \)
Step 3: Now, find the average of the estimated value from Step 1 (3) and the quotient from Step 2 (18.6667).
\( \frac{3 + 18.6667}{2} = 10.8333 \)
Step 4: Repeat Step 2 and Step 3 with the average from Step 3 (10.8333) until you have the desired precision.
\( \frac{56}{10.8333} = 5.1631 \)
\( \frac{10.8333 + 5.1631}{2} = 7.9982 \)
Step 5: Continue repeating Step 2 and Step 3 until you reach the desired precision. Let's repeat Step 2 and Step 3 one more time:
\( \frac{56}{7.9982} = 7.0008 \)
\( \frac{7.9982 + 7.0008}{2} = 7.4995 \)
So, the cubic root of 56 (to four decimal places) is approximately 7.4995.
To find the cubic root of a number, you can use a scientific calculator or perform the calculation manually by applying some mathematical principles. Let's break down the steps:
Step 1: Start with an initial guess. In this case, we can start with a guess of 4.
Step 2: Cube the initial guess. 4^3 = 64.
Step 3: Compare the result to the target number (56 in this case). Since 64 is greater than 56, we need to reduce our guess.
Step 4: Adjust the guess based on the comparison. We can try a smaller guess, like 3.
Step 5: Cube the new guess. 3^3 = 27.
Step 6: Compare the result to the target number (56). Since 27 is less than 56, we need to increase our guess.
Step 7: Adjust the guess again, this time slightly higher. Let's try 3.5.
Step 8: Cube the new guess. 3.5^3 = 42.875.
Step 9: Compare the result to the target number (56). Since 42.875 is less than 56, we need to increase our guess.
Step 10: Repeat steps 7-9, gradually refining the guess until we get close enough to the target number.
By following these steps, you'll eventually find that the cubic root of 56 (rounded to three decimal places) is approximately 3.779. Note that this is an approximate value, as the exact cubic root of 56 would be an irrational number.