Problem 65 - Elementary Algebra

Question

Use the order of operations to simplify the quantities for the following problems. $$ \frac{5\left[8^{2}-9(6)\right]}{2^{5}-7}+\frac{7^{2}-4^{2}}{2^{4}-5} $$

Step-by-Step Solution

Answer

Question: Simplify the given expression $$5[8^{2} - 9(6)] + \frac{7^{2} - 4^{2}}{2^{4} - 5}$$ Answer: 5

Step 1: Calculate the operations inside the parentheses

First, we need to solve the operations inside the parentheses. Let's start with the first fraction: $$ 5\left[8^{2}-9(6)\right] $$ Now, carry out the exponent and multiplication operations: $$ 5\left[64-54\right] $$ Continue to solve inside the bracket: $$ 5[10] $$ Now, for the second fraction: $$ \frac{7^{2}-4^{2}}{2^{4}-5} $$ Solve the exponents: $$ \frac{49-16}{16-5} $$

Step 2: Simplify the fractions

Let's solve the numerators and denominators for the two fractions: For the first fraction, multiply by 5: $$ \frac{50}{\textrm{Since we have not yet calculated the denominator}} $$ Now, for the second fraction, solve the numerators and denominators: $$ \frac{33}{11} $$

Step 3: Add the two fractions together

Since the denominators of the first and second fractions are different, we will first need to find the value for the denominator of the first fraction (recall that we left the denominator calculation in step 1): $$ {2^{5}-7} $$ Calculate the exponent and solve for the denominator: $$ {32-7} $$ $$ {25} $$ Now, we can write the first fraction with the calculated denominator: $$ \frac{50}{25} $$ Simplify the first fraction: $$ 2 $$ Now, we can add the two simplified fractions (2 and 33/11) together: $$ 2+\frac{33}{11} $$ The second fraction can be simplified as: $$ \frac{33}{11} = 3 $$ Finally, add the simplified fractions together: $$ 2+3 $$ The simplified expression is: $$ 5 $$

Key Concepts

Simplifying Expressions

Simplifying expressions is a crucial math skill that involves performing operations in a step-by-step manner to make an equation or expression as simple as possible. The key to simplifying effectively is understanding the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition, and Subtraction). By consistently applying these rules, you can ensure that you simplify expressions correctly. In our problem, we started by simplifying individual fractions, focusing first on operations inside parentheses, followed by dealing with exponents and multiplication. Once the individual fractions were simplified, final additions were performed, leading to the simplified result.

Fractions

Fractions represent a part of a whole and can sometimes look intimidating to simplify, especially when they involve multiple operations. In this exercise, each fraction required separate consideration of its numerator and denominator.

Here’s what to look for when simplifying fractions:

• Calculate operations within the numerator and denominator separately.
• Pay attention to the order of operations within each fraction.
• Once the operations are complete, check if the fraction can be reduced by finding common factors in the numerator and denominator.

Reducing fractions is vital to ensure they are at their simplest form, making subsequent calculations more manageable.

Exponents

Exponents are another layer of complexity in expressions and involve raising a number to a specified power. They are second in line after parentheses according to the order of operations. In our exercise, several numbers had exponents, like \(8^2\) and \(7^2\).

Here's how to handle them:

• Calculate any exponents as early as possible as they're not enclosed in parentheses.
• Remember that exponents indicate repeated multiplication, e.g., \(8^2\) means \(8 \times 8\).
• Avoid errors by double-checking that you've correctly calculated each power of numbers before proceeding with further simplifications.

Understanding and simplifying exponents step by step can significantly affect the accuracy and ease of solving expressions.

Multiplication and Division

These operations are treated with equal priority in the order of operations, and they come after dealing with exponents. In some cases, you might have to do multiplication first, followed by division, depending on their order of appearance from left to right in the expression. In the exercise, multiplication was used to solve \(9(6)\) and \(5[10]\) after all the exponents had been simplified.

Key points include:

• Combine multiplication and division steps without interruption unless high-priority operations (parentheses, exponents) direct otherwise.
• Work from left to right if both operations appear at the same level.
• Ensure that each part of the expression is resolved before moving on to subsequent steps.

By carefully executing multiplication and division, the expression simplifies progressively, laying ground for final simplification steps like addition or subtraction.