Problem 63 - Fundamentals of Mathematics

Question

Find each value. Check each result with a calculator. $$\frac{6^{3}-2 \cdot 10^{2}}{2^{2}}+\frac{18\left(2^{3}+7^{2}\right)}{2(19)-3^{3}}$$

Step-by-Step Solution

Answer

The value is approximately 97.27.

Step 1: Calculate Individual Powers

First, compute the individual powers: \(6^3\), \(10^2\), \(2^2\), and \(2^3\), and \(7^2\). \(6^3 = 216\), \(10^2 = 100\), \(2^2 = 4\), \(2^3 = 8\), \(7^2 = 49\).

Step 2: Substitute and Simplify Numerator

Substitute the calculated powers into the expression: \(\frac{6^{3}-2 \cdot 10^{2}}{2^{2}}\) becomes \(\frac{216 - 2 \cdot 100}{4}\). Simplify the numerator: \(216 - 200 = 16\).

Step 3: Calculate First Fraction

Divide the simplified numerator by the denominator: \(\frac{16}{4} = 4\).

Step 4: Simplify Other Parts of the Expression

For the second part, substitute in the calculations from Step 1: \(18(2^3 + 7^2)\) becomes \(18(8 + 49) = 18 \times 57\).Compute \(2 \times 19 - 3^3\):\(2 \times 19 = 38\) and \(3^3 = 27\), so \(38 - 27 = 11\).

Step 5: Calculate Second Fraction

Divide the second simplified expression: \(\frac{18 \times 57}{11}\). First find \(18 \times 57\): \(18 \times 57 = 1026\). Now divide: \(\frac{1026}{11} = 93.27\approx 93.27\).

Step 6: Add the Two Results

Add the results of the two fractions together: \(4 + 93.27 = 97.27\).

Step 7: Double Check Using Calculator

Use a calculator to verify each step and ensure the accuracy of the computed values and operations, especially the division and multiplication. Ensure rounding, if any, has been accounted for.

Key Concepts

Powers and Exponents

Powers and exponents are essential in mathematics to represent repeated multiplication of a number by itself. When you see a number with a superscript, like \(6^3\), this indicates that the number 6 is multiplied by itself three times. Calculating powers involves determining the base (the number being multiplied) and the exponent (the number of times it is multiplied).

• Example: \(6^3 = 6 \times 6 \times 6 = 216\).
• It's crucial to understand that multiplying by zero or one results in special cases where the power results in the base or zero respectively.

When solving expressions, it's important to compute powers first as guided by the order of operations (PEMDAS: Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). Always start by solving any exponents in the expression before moving on to other operations. This step can simplify your calculations significantly and ensure accuracy.

Fractions

Fractions represent parts of a whole and are composed of a numerator and a denominator. The numerator is the top number indicating how many parts are being considered, while the denominator is the bottom number indicating the total number of parts.

• Example: In \(\frac{16}{4}\), 16 is the numerator, and 4 is the denominator.
• Fractions can be simplified by dividing both the numerator and the denominator by their greatest common divisor (GCD).

When dealing with complex expressions containing fractions, like the original exercise, always remember to compute the expressions within the numerator and denominators separately first. Once simplified, you can divide the numerator by the denominator to get a single value. This method helps make calculations more manageable and reduces the potential for errors.

Simplification

Simplification is the process of rewriting a more complex expression in a simpler or more efficient form. This often involves reducing fractions or solving arithmetic calculations within an expression. In our original exercise, simplification began by resolving the most straightforward parts, such as calculating the powers and substituting them back into the expression.

• Simplifying also includes combining like terms or eliminating extra parentheses when nothing inside changes the outcome.
• This reduces clutter in expressions and makes it easier to execute further calculations accurately.

By simplifying expressions as much as possible, you make subsequent steps, like addition or multiplication, more straightforward, thereby minimizing the possibility of mistakes. It allows for a cleaner solution and a more precise path to the answer.

Numerical Calculations

Numerical calculations involve performing arithmetic operations such as addition, subtraction, multiplication, or division. It's essential to be confident with these calculations as they form the basis of solving most math problems. In our expression, once individual powers and fractions were calculated, basic numerical calculations were necessary to reach the final result.

• Start with multiplication and division before moving on to addition and subtraction.
• Always double-check your calculations to ensure accuracy, especially with complex numbers or large expressions.

For example, after simplifying individual parts of the original expression, the final numerical task was to add the results of the two fractions, yielding the end solution of 97.27. Ensuring each numerical calculation is correct is crucial, as even a small error can lead to a wrong outcome.