Question

Use the Chinese cube root algorithm to find the cube root. of \(12,812,904\).

Short answer

Answer: The cube root of 12,812,904 is 221.

Step-by-step solution

Group the digits in triplets

In the given number, \(12,812,904\), we group the digits in triplets starting from the right side: $$12,\ 812,\ 904$$

Determine the largest cubic number

We need to find the cubic number \(A^3\) such that \(A^3 \leq 12\). The largest cubic number less than or equal to 12 is \(2^3 = 8\). So, \(A = 2\) is our first digit of the cube root.

Subtract the cubic number

Now, subtract the cubic number (\(2^3 = 8)\) from the leftmost triplet (12): $$12 - 8 = 4$$

Bring down the next grouped number

Bring down the next grouped number (812) and form a dividend with the remainder from the previous step: $$4,\ 812$$

Find an appropriate dividend

We need to find a new number, \(N\), such that \(N(2^3 * 2 + N) \leq 4,\ 812\). Testing \(N = 2\), we get: $$2(8 * 2 + 2) = 2(18) = 36$$ Since \(36\) is less than or equal to \(4,812\), \(N = 2\) works. The second digit of the cube root is 2.

Subtract the cubic number times the new number

Subtract \(2^3 * 2 * 2\) from the new dividend: $$4,\ 812 - 2(18) = 4,\ 812 - 36 = 4,\ 776$$

Repeat the previous steps

Now repeat steps 4 to 6: 1. Bring down the last grouped number (904) and form a dividend with the remainder from the previous step: $$4,\ 776,\ 904$$ 2. Find a new number, \(M\), such that \(M(2^3 * 22 + M) \leq 4,\ 776,\ 904\). Testing \(M = 3\), we get: $$3(1928) = 5,\ 784$$ Since \(5,\ 784\) is greater than \(4,\ 776,\ 904\), we reduce \(M\) to 2: $$2(2^3 * 22 + 2) = 2(1928) = 3,856$$ 3. Subtract the cubic number times the new number: $$4,\ 776,\ 904 - 3,856 = 4,\ 773,\ 048$$ Since we have no more triplets to bring down, our final cube root will have 22 as the first two digits. So the cube root is \(22M\), testing with \(M=1\), we get \(221^3=10,792,461\) which is less than \(12,812,904\), testing with \(M=2\), we get \(222^3=10,984,832\) which is not less than \(12,812,904\), hence the cube root of \(12,812,904\) is \(\boxed{221}\).

Key concepts

Mathematics Education

Understanding mathematical concepts deeply is crucial to education in mathematics. Often, students encounter difficulty with the abstraction in math, which can be mitigated through the exploration of practical methods used historically.

The Chinese cube root algorithm exemplifies a step-by-step approach to solving cubic roots, which can be a challenging topic. By dissecting complex processes into manageable steps, educators can aid students in building their problem-solving skills progressively. This not only helps in grasping specific methods such as the Chinese cube root algorithm but also instills a broader sense of algorithmic thinking, reinforcing students' confidence in tackling various mathematical challenges.

Cube Root Calculation

Computing cube roots might seem daunting without the aid of a calculator, but algorithms like the Chinese cube root algorithm offer an intuitive approach. To grasp this method, one must understand the relationship between numbers and their cubic powers.

In our exercise, breaking down the number into triplets and systematically finding the largest cubic number less than or equal to the first group simplifies what initially appears to be a complex calculation. The step-by-step process of subtraction and bringing down the next group encourages a thorough comprehension of how cubic numbers are constructed and deconstructed, an important skill in mathematics.

Algorithmic Problem Solving

The beauty of algorithms lies in their systematic procedure for solving problems. These stepwise instructions encourage logical reasoning and structured thinking.

The Chinese cube root algorithm is a prime example, requiring the solver to follow a defined sequence of actions: grouping digits, determining cubic numbers, subtracting, and bringing down the next group. By applying this algorithm to find cube roots, students learn to break down problems into solvable units, a valuable strategy in all areas of mathematics and in real-world problem-solving scenarios.

Historical Mathematical Methods

Exposure to historical mathematical methods like the Chinese cube root algorithm gives context to the development of mathematical thought and highlights the creativity of past mathematicians.

These methods provide unique insights into how different cultures approached mathematical problems before the advent of modern technology. Understanding such algorithms helps to appreciate the evolution of mathematical knowledge and its cultural significance. Moreover, it can inspire students to see mathematics as a dynamic and evolving field, imbued with human history and innovation.