Problem 19 - Fundamentals of Mathematics

Question

Determine the value of each of the following. \(\frac{3^{3}+2^{3}}{6^{2}-29}+5\left(\frac{8^{2}+2^{4}}{7^{2}-3^{2}}\right) \div \frac{8 \cdot 3+1^{8}}{2^{3}-3}\)

Step-by-Step Solution

Answer

The final value is 7.

Step 1: Simplify Each Component

Start by calculating each part of the expression separately.- Calculate \(3^3 + 2^3\): \[3^3 = 27, \quad 2^3 = 8\] Therefore, \(3^3 + 2^3 = 27 + 8 = 35\).- Calculate \(6^2 - 29\): \[6^2 = 36\] Therefore, \(6^2 - 29 = 36 - 29 = 7\).- For \(\frac{3^3 + 2^3}{6^2 - 29}\), substitute the calculated values: \[\frac{35}{7} = 5\].Next:- Calculate \(8^2 + 2^4\): \[8^2 = 64, \quad 2^4 = 16\] Therefore, \(8^2 + 2^4 = 64 + 16 = 80\).- Calculate \(7^2 - 3^2\): \[7^2 = 49, \quad 3^2 = 9\] Therefore, \(7^2 - 3^2 = 49 - 9 = 40\).- For \(\frac{8^2 + 2^4}{7^2 - 3^2}\), substitute the calculated values: \[\frac{80}{40} = 2\].

Step 2: Calculate Entire Expression

Substitute the simplified components back into the expression. Now evaluate:\[5 + 5 \times 2 \div \frac{8 \cdot 3 + 1^8}{2^3 - 3}\]- Calculate \(8 \cdot 3 + 1^8\): \[8 \cdot 3 = 24, \quad 1^8 = 1\] Therefore, \(8 \cdot 3 + 1 = 25\).- Calculate \(2^3 - 3\): \[2^3 = 8\] Therefore, \(8 - 3 = 5\).Thus, \(\frac{8 \cdot 3 + 1}{2^3 - 3} = \frac{25}{5} = 5\).

Step 3: Perform Final Operations

Substitute into the simplified expression:\[5 + 5 \times 2 \div 5\]- Calculate the division and multiplication: \(5 \times 2 = 10\) Then,\(\frac{10}{5} = 2\).- Add the simplification result to the initial number: \[5 + 2 = 7\].

Key Concepts

Exponents

Exponents are a way of expressing repeated multiplication of a number by itself. When you see a number raised to a power, it means that the base number is multiplied by itself as many times as the exponent suggests. For example,

• If you have \(3^3\), it means you multiply 3 by itself three times: \(3 \times 3 \times 3 = 27\).
• Similarly, with \(2^3\), you multiply 2 by itself three times: \(2 \times 2 \times 2 = 8\).

Exponents are incredibly useful in simplifying large mathematical expressions and are often one of the first operations performed when following the order of operations. Remember that squaring a number, as in \(6^2\), means multiplying the number by itself just two times, which in this example gives 36.

Fractions

Fractions represent a part of a whole. They consist of a numerator over a denominator, separated by a line. Understanding fractions is essential as they often appear in mathematical expressions.

• A fraction like \(\frac{35}{7}\) involves dividing the numerator (35) by the denominator (7), which gives 5.
• Similarly, \(\frac{80}{40}\) simplifies to 2 by dividing 80 by 40.

When you encounter fractions, it’s helpful to simplify them as much as possible to make calculations easier later on. The process of simplifying fractions can sometimes seem like magic because it reduces the expression without changing its value, making it more manageable.

Simplification

Simplification is a core process in solving mathematical problems, especially complex ones. It involves reducing expressions to their simplest form, making them easier to work with.
For instance, when given a complex expression, you can start by simplifying each individual component, as we did with \(3^3 + 2^3\) and \(6^2 - 29\). Performing these operations separately can make the overall problem less intimidating.
Simplifying fractions and understanding how to combine like terms or reduce expressions are essential skills. Simplification also helps in minimizing potential errors and ensures that calculations are as clear and straightforward as possible.

Division and Multiplication Operations

Understanding division and multiplication is crucial when solving mathematical expressions. These operations are often performed after simplifying exponents and fractions.

• In the expression \(5 + 5 \times 2 \div 5\), multiplication and division are performed first, from left to right.
• While division, as seen in \(\frac{10}{5} = 2\), reduces the fraction to a simpler form.

Remember that when both division and multiplication appear in an expression, they should be performed in the order they occur, from left to right, as per the order of operations. This ensures accuracy and consistency in problem-solving.