Problem 66 - Elementary Algebra

Question

Use the order of operations to simplify the quantities for the following problems. $$ \frac{(2+1)^{3}+2^{3}+1^{3}}{6^{2}}-\frac{15^{2}-[2(5)]^{2}}{5 \cdot 5^{2}} $$

Step-by-Step Solution

Answer

Question: Simplify the following expression using the order of operations: $$ \frac{(2+1)^{3}+2^{3}+1^{3}}{6^{2}}-\frac{15^{2}-[2(5)]^{2}}{5\cdot 5^{2}} $$ Answer: 0

Step 1: Solve the expressions inside the brackets or parenthesis

First, we need to solve the expressions inside the brackets. There are two brackets in the expression: 1. (2+1) in the numerator of the first fraction and 2. [2(5)] in the numerator of the second fraction. Let's solve them: 1. \((2+1)=3\) 2. \([2(5)]=10\) Now the expression becomes: $$ \frac{(3)^{3}+2^{3}+1^{3}}{6^{2}}-\frac{15^{2}-10^{2}}{5\cdot 5^{2}} $$

Step 2: Solve the exponents

Next, we need to solve the exponents. The exponents in the expression are: 1. \((3)^{3}\), 2. \(2^{3}\), 3. \(1^{3}\), 4. \(6^{2}\), in the second fraction (denominator), 5. \(15^{2}\), and 6. \(5^{2}\), in the second fraction (denominator). Let's solve these exponents: 1. \((3)^{3}=27\), 2. \(2^{3}=8\), 3. \(1^{3}=1\), 4. \(6^{2}=36\), 5. \(15^{2}=225\), and 6. \(5^{2}=25\). Now the expression becomes: $$ \frac{27+8+1}{36}-\frac{225-10^{2}}{5\cdot 25} $$

Step 3: Solve addition and subtraction in the numerators

Now, we will solve the addition and subtraction operations in the numerators of both fractions: 1. \(27+8+1 = 36\) 2. \(225-10^{2} = 225-100 = 125\) Now the expression becomes: $$ \frac{36}{36}-\frac{125}{5\cdot 25} $$

Step 4: Solve multiplication in the denominator

Next, we'll solve the multiplication operation in the denominator of the second fraction: \(5\cdot 25 = 125\) Now the expression becomes: $$ \frac{36}{36}-\frac{125}{125} $$

Step 5: Solve division in fractions

Let's now perform the division operations for both fractions: 1. \(\frac{36}{36} = 1\) 2. \(\frac{125}{125} = 1\) Now the expression becomes: $$ 1-1 $$

Step 6: Solve subtraction

Finally, we solve the subtraction: \(1-1 = 0\) Thus, the simplified expression is: $$ 0 $$

Key Concepts

Exponents

Exponents are a crucial part of mathematical expressions, particularly when simplifying equations. An exponent indicates how many times a number, known as the base, is multiplied by itself. For instance, in the expression \( 3^3 \), the number 3 is raised to the power of 3, which means \( 3 \times 3 \times 3 = 27 \).

Understanding exponents is vital for simplifying expressions. They must be calculated before tackling addition or subtraction. Your problem had several exponents:

• \((3)^3\) which simplifies to 27
• \(2^3\) simplifies to 8
• \(1^3\) simplifies to 1
• \(6^2\) simplifies to 36
• \(15^2\) simplifies to 225
• \(5^2\) simplifies to 25

Each of these calculations reduces the complexity of the original problem, making it simpler to continue solving the rest of the expression.

Fractions

Fractions represent a way to show division in mathematical notation. They contain a numerator and a denominator. Here, the numerator is divided by the denominator, which means that fractions are tackled in the order of operations after exponents and before addition or subtraction.

In your exercise, fractions were used extensively. First, parts of the expression inside the brackets simplified to: \( \frac{36}{36} \) and \( \frac{125}{5 \cdot 25} \).

For simplification, calculate any remaining operations inside the fraction itself:

• Multiplication in the denominator: \(5 \cdot 25 = 125\).
• Division: \( \frac{125}{125} = 1 \).

This simplification is crucial in reaching the ultimate goal, which is transforming these fractions into their simplest form before moving on to subtraction.

Brackets and Parentheses

Using brackets and parentheses helps dictate the order in which operations should be performed in mathematics, ensuring clarity and precision. They indicate which parts of an expression should be simplified first, overriding the typical order of operations.

In your example, the expression required evaluating the numbers in brackets and parentheses before doing anything else:

• Inside parentheses: \((2 + 1) = 3\).
• Inside brackets: \([2(5)] = 10\).

This approach ensures that components inside the brackets and parentheses are fully resolved before proceeding with the rest of the calculation, maintaining an organized and accurate process.

Simplify Expressions

Simplifying expressions involves following a systematic order: resolving operations inside parentheses and brackets, calculating exponents, handling any multiplication or division, and finally performing addition or subtraction.

After decoding operations in brackets, you tackle exponents. Then it's time to solve operations inside numerators and denominators of fractions, followed by simplifying the fractions themselves:

• Addition in the numerator: \(27 + 8 + 1 = 36\).
• Subtraction in the other numerator: \(225 - 100 = 125\).
• Fractions simplify: \( \frac{36}{36} = 1 \) and \( \frac{125}{125} = 1 \).

Finally, perform any remaining operations, such as subtracting the simplified expressions, obtaining the solution \( 0 \). By following this chain of steps, expressions become significantly simpler to manage.