Problem 68 - Fundamentals of Mathematics

Question

Find the roots (using your knowledge of multiplication). Use a calculator to check each result. \(\sqrt[3]{27}\)

Step-by-Step Solution

Answer

The cube root of 27 is 3.

Step 1: Understanding Cube Roots

The cube root of a number is a value that, when multiplied by itself three times, gives the original number. For example, if a number \( x \) is a cube root of 27, then \( x^3 = 27 \).

Step 2: Estimation and Testing

Since we know that \( 3 \times 3 \times 3 = 27 \), we can estimate that the cube root of 27 is 3. This means \( \sqrt[3]{27} = 3 \).

Step 3: Verification Using a Calculator

To verify this solution, use a calculator to find the cube root of 27. Enter 27, use the cube root function if available, or compute 27 raised to the power of \( \frac{1}{3} \), which should confirm that \( \sqrt[3]{27} = 3 \). The calculator should display 3, confirming our result.

Key Concepts

Multiplication

Multiplication is a core mathematical operation used in various contexts, including finding cube roots. To understand the cube root, we first need to recall that multiplication involves combining equal groups. For cube roots, you multiply the root by itself three times. For example, with the cube root of 27, we try to find a number which when multiplied three times (or cubed) leads back to 27.
In our example, 3 is the number we multiply repeatedly:

• First 3: 3
• Second 3: 3 \( \times \) 3 = 9
• Third 3: 3 \( \times \) 9 = 27

Therefore, the cube root of 27 ( \( \sqrt[3]{27} \) ) is 3. Multiplication in this context helps reinforce the concept by demonstrating the “3 times” attribute of cube roots, ensuring a solid comprehension of the operation involved.

Estimation

Estimation is a handy mathematical technique where exact answers aren’t required immediately. Instead, you make an educated guess, which simplifies the problem or provides a closer starting point.
When dealing with cube roots, estimating can involve identifying close perfect cubes around the number. For instance, estimating the cube root of 27 starts by recognizing that:

• The cube of 2 is 8 (2 \( \times \) 2 \( \times \) 2).
• The cube of 3 is 27 (3 \( \times \) 3 \( \times \) 3).
• This tells us 27 itself is a perfect cube of 3.

Through estimation, you narrow down potential answers significantly, turning what could seem like guesswork into an almost certain solution. This step is invaluable when approaching more complex numbers, where exact calculations may initially seem overwhelming.

Calculator Use

Using a calculator effectively can streamline finding cube roots, especially for larger or non-perfect cubes. To check if your estimated cube root is correct, follow these simple steps:

• Enter the number (in this case, 27) into the calculator.
• Use the cube root function, often depicted as \( \sqrt[3]{x} \).
• If the calculator doesn’t have a cube root button, calculate 27 raised to the power of \( \frac{1}{3} \).
• The correct outcome should match the estimated result.

In this exercise, calculating \( \sqrt[3]{27} \) on a calculator should give 3, confirming the manual steps. Calculators make dealing with intricate numbers more feasible and provide a means to double-check manual calculations, thereby reducing errors.

Mathematical Verification

Mathematical verification is the process of confirming that a calculated result is accurate. Verifying your answer is crucial in problem-solving to ensure the output is reliable. Once you have an approximate cube root, you must confirm this result using another method, such as a calculator or revisiting multiplication.
In this instance, the cube root of 27 is postulated as 3.

• Check using multiplication: \( 3 \times 3 \times 3 = 27 \).
• Use a calculator to ensure the cube root of 27 matches the estimated value.
• Adjust if results deviate; errors could arise from miscalculations or misinterpretation.

Through verification, you double-check your solution’s authenticity—emphasizing accurate steps enhance confidence in your mathematical abilities. A validated result confirms understanding and mastery of calculating cube roots.