Problem 33 - Fundamentals of Mathematics

Question

Find each value. Check each result with a calculator. \(\sqrt[3]{8}+8-2 \cdot 5\)

Step-by-Step Solution

Answer

The result is 0.

Step 1: Calculate the Cube Root

First, find the cube root of 8. The cube root of a number is a value that, when multiplied by itself three times, equals that number. Here, \( \sqrt[3]{8} = 2 \) since \( 2 \times 2 \times 2 = 8 \).

Step 2: Simplify the Multiplication

Next, calculate \( 2 \cdot 5 \). Multiply the two numbers together: \( 2 \cdot 5 = 10 \).

Step 3: Perform the Operations

Now substitute the results from Steps 1 and 2 back into the original expression: \( 2 + 8 - 10 \).

Step 4: Calculate Final Result

Add and subtract the numbers: \( 2 + 8 = 10 \) and \( 10 - 10 = 0 \). So, the result is 0.

Step 5: Verify with Calculator

Use a calculator to confirm the calculation: \( \sqrt[3]{8} = 2 \), \( 2 \cdot 5 = 10 \), and \( 2 + 8 - 10 = 0 \). The calculator will confirm that the steps and arithmetic were done correctly.

Key Concepts

Cube Root

In mathematics, the cube root is an essential concept. It refers to a number which, when used as a factor three times, gives the original number. For instance, picking out the cube root of 8 means finding a number that multiplies itself three times to give the result of 8. Here, the cube root of 8 is indeed 2 because when 2 is multiplied by itself three times, it results in 8. In notation, the cube root is represented as \( \sqrt[3]{a} \) where \( a \) is the number you want to find the cube root of. To calculate it manually or mentally, you often rely on memorizing cube root values of small numbers or use a calculator for convenience.Understanding cube roots is crucial as it lays the groundwork for solving more complex problems involving roots in algebra. Knowing this concept helps in evaluating expressions and in solving equations where variables might be raised to the third power.

Order of Operations

The order of operations is a fundamental principle in mathematics used to solve expressions correctly. It determines the sequence in which operations such as addition, subtraction, multiplication, and division are performed. Usually, this concept is remembered with the acronym PEMDAS:

• Parentheses - solve expressions in parentheses first.
• Exponents - resolve exponents next, including roots.
• Multiplication and Division - take care of these operations next, from left to right.
• Addition and Subtraction - complete these operations last, from left to right.

In our case, we first solved the cube root (considered as the exponent in the acronym), then proceeded with multiplication and finally performed the addition and subtraction. This approach ensures that the mathematical operations yield the correct result, no matter how complex the expression becomes.

Verification with Calculator

To ensure that our manual calculations are correct, verifying the result with a calculator is a beneficial step. Calculators are excellent tools for checking arithmetic operations and provide an immediate confirmation for numerical problems. In dealing with our expression, after resolving the operations manually, inputting the expression into the calculator as \( \sqrt[3]{8} + 8 - 2 \cdot 5 \) directly helps verify our result. The calculator confirms each intermediary step: the cube root calculation of 2, the multiplication result of 10, and the final arithmetic leading to 0. Using a calculator not only assures the accuracy of our solution but also helps in understanding the sequence of operations more clearly. This approach confirms one’s understanding and provides a reliable self-check method for arithmetic expressions.