How to Calculate the Cube Root of 674563: A Step-by-Step Guide

Calculating the cube root of 674563 can be done manually using the long division method with the binomial formula. This article provides a detailed step-by-step guide on how to find the cube root of 674563 by hand, without the use of a calculator. We'll walk you through each step, including accessing the cube root using a calculator and logarithmic methods. Let's dive in!

Method: Long Division with Binomial Formula

To find the cube root of 674563 manually, follow these steps:

Step 1: Group the Digits

First, group the digits of the number into sets of three starting from the right. For 674563, the digits are grouped as 674 and 563.

Step 2: Find the Initial Cube Root

Identify the largest perfect cube less than or equal to the first group of digits (674). The largest perfect cube less than 674 is 512, which is (8^3). So, the first digit of the cube root is 8.

Step 3: Calculate the Remainder

Subtract 512 from 674 to get the remainder:

674 - 512 162

Now, bring down the next three digits (563) to form the next dividend:

162563

Step 4: Apply the Binomial Formula

Use the binomial formula to find the next digit of the cube root:

Let (a 8) (the current digit of the cube root) and (b) (the next digit of the cube root). The formula is:

300a^2b 30ab^2 b^3

Substitute (a 8) and find the largest possible (b) such that the result is less than or equal to 162563:

1. Try (b 7):

300(8^2)(7) 30(8)(7^2) 7^3 146503

146503 is less than 162563, so (b 7) is valid.

2. Calculate the new remainder:

162563 - 146503 16060

Bring down the next three digits (000) to form the next dividend:

16060000

Continue the process to find the next digit after the decimal point, and so on.

Step 5: Verify the Cube Root

The cube root of 674563 is approximately 87.7, which can be verified by calculating:

87.7 x 87.7 x 87.7 674526.133

Using a Calculator

If you prefer a faster method, you can use a calculator to find the cube root directly:

Use the exponent function (x1/3): Enter 6745631/3 Press the equals button: Result: 87.70159775

Alternatively, you can use the cube root button, which is often denoted as (sqrt[3]{x}).

Conclusion

Calculating the cube root of 674563 using the long division method with the binomial formula is a time-consuming process but educational. For practical use, a calculator is the most efficient method. Regardless of the method, the cube root of 674563 is approximately 87.7.