Problem 71 - Fundamentals of Mathematics

Question

Find the roots (using your knowledge of multiplication). Use a calculator to check each result. \(\sqrt[3]{216}\)

Step-by-Step Solution

Answer

The cube root of 216 is 6.

Step 1: Understanding the Cube Root

The problem asks for the cube root of 216, denoted as \( \sqrt[3]{216} \). A cube root of a number is a value that, when multiplied by itself three times, gives the original number.

Step 2: Identifying the Possible Root

To find \( \sqrt[3]{216} \), you need to consider what number, when multiplied by itself three times, equals 216. Since multiplication is involved, think of smaller whole numbers and increase gradually if needed.

Step 3: Testing Possible Values

Start testing with smaller numbers: - If \( 5 \times 5 \times 5 = 125 \), clearly too small.- If \( 6 \times 6 \times 6 = 216 \), this matches the target.

Step 4: Verifying Using Multiplication

Verify by multiplying: \( 6 \times 6 = 36 \)\( 36 \times 6 = 216 \). So, \( 6^3 = 216 \), confirming that 6 is indeed the cube root of 216.

Step 5: Checking with a Calculator

Use a calculator and verify. Enter \( 6^3 \) to ensure it equals 216. Alternatively, calculate \( \sqrt[3]{216} \) directly to confirm it equals 6.

Key Concepts

Roots of Numbers

Understanding the root of a number, especially cube roots, is key to mastering many math problems. For example, the cube root of 216, written as \( \sqrt[3]{216} \), is the number that, when multiplied by itself three times, results in 216. This concept is crucial for unlocking various mathematical puzzles and equations involving powers.

Finding the cube root involves a bit of detective work. You'll want to think of numbers that are potential candidates. Begin with small whole numbers and test by multiplying them by themselves three times. As you become more comfortable with this process, you'll be able to estimate and locate the roots quicker.

• The cube root of 8 is 2 because \( 2 \times 2 \times 2 = 8 \).
• The cube root of 27 is 3, since \( 3 \times 3 \times 3 = 27 \).
• And, as in our example, the cube root of 216 is 6 because \( 6 \times 6 \times 6 = 216 \).

When you find the cube root, you're solving a mini-puzzle, discovering the number that unravels the mystery of the cube.

Multiplication Verification

Verifying your solution through multiplication is an essential step. It ensures accuracy in finding roots. Let’s say we guessed that the cube root of 216 was 6. To verify, you would multiply as follows: perform \( 6 \times 6 = 36 \), then take that product and multiply it by 6 again, \( 36 \times 6 = 216 \).

This process solidifies your understanding and confidence in your answers. You can use this approach for all roots:

• Square Root Check: For example, check if \( \sqrt{25} = 5 \) by verifying \( 5 \times 5 = 25 \).
• Cube Root Check: Verify that \( 6^3 = 216 \) for the cube root of 216.

Practicing this method regularly will enhance your multiplication skills and provide peace of mind that your answers are correct.

Using Calculator for Math

Using a calculator is like having a math buddy to double-check your answers. When tackling cube roots like \( \sqrt[3]{216} \), you can use a calculator in two main ways:

• Direct Input: Simply input \( \sqrt[3]{216} \) into the calculator to see the result is 6.
• Verification: After guessing a root, like 6, input \( 6^3 \) to confirm you get 216.

Calculators are excellent for saving time and reducing errors. They relieve the mental load while allowing you to focus on understanding the process. Often, seeing the correct number appear will help reinforce the logic you used in your head when finding the root.

Remember, while calculators are helpful, it's essential to still grasp the fundamental concepts behind the math, ensuring you can solve problems both with and without a calculator.