Estimating Cube Root of 12167

To estimate the cube root of 12167, we can use a method called the estimation method. This method involves finding the closest perfect cube to the given number and then using that perfect cube as a reference point to estimate the cube root.

Step-by-Step Process

1. Find the closest perfect cube to 12167.

We know that the cube of 23 is 12167, so 23 is the closest perfect cube to 12167.

2. Use the perfect cube as a reference point to estimate the cube root.

To estimate the cube root of 12167, we can use the fact that:

- If x is the cube root of a number, then x^3 is that number.

- If a and b are two numbers such that a < b,="" then="" the="" cube="" root="" of="" a="" is="" less="" than="" the="" cube="" root="" of="" />

Using these facts, we can estimate the cube root of 12167 as follows:

- Since 23^3 = 12167, the cube root of 12167 is approximately 23.

- To refine our estimate, we can use the fact that:

- If x is an approximation of the cube root of a number, then x^3 is an approximation of that number.

- If a and b are two numbers such that a < b,="" then="" (a="" +="" b)/2="" is="" a="" number="" between="" a="" and="" />

Using these facts, we can refine our estimate as follows:

- We can find a number between 12167 and 23^3 by taking the average of 12167 and 23^3, which is (12167 + 23^3)/2 = 12245.

- We can then find the closest perfect cube to 12245, which is 12^3 = 1728.

- Since 1728 is less than 12245, we know that the cube root of 12167 is between 23 and 12.

- We can then repeat the process, taking the average of 23 and 12 to get 17.5, finding the closest perfect cube to 17.5, which is 8^3 = 512, and repeating until we get a desired level of accuracy.

Final Answer

Using the estimation method, we can estimate the cube root of 12167 as follows:

- The cube root of 12167 is approximately 23.

- Refining our estimate using the average of 12167 and 23^3, we get 12245.

- Finding the closest perfect cube to 12245, we get 1728.

- Taking the average of 23 and 12, we get 17.5.

- Finding the closest perfect cube to 17.5, we get 512.

- Refining further, we get 23.5 as our final estimate.

- Therefore, the cube root of 12167 is approximately 23.5.

23