Problem 68 - Elementary Algebra

Question

For exercises 1-80, evaluate. $$ \frac{(10-1)^{2}-81}{2^{3}} $$

Step-by-Step Solution

Answer

0

Step 1 - Simplify the Numerator

First, simplify the expression inside the parentheses: \[ (10-1) = 9 \]

Step 2 - Square the Result

Next, square the result from Step 1: \[ 9^2 = 81 \]

Step 3 - Subtract 81

Subtract 81 from the squared value: \[ 81 - 81 = 0 \]

Step 4 - Simplify the Denominator

Simplify the denominator: \[ 2^3 = 8 \]

Step 5 - Divide the Numerator by the Denominator

Finally, divide the simplified numerator by the simplified denominator: \[ \frac{0}{8} = 0 \]

Key Concepts

Simplifying Expressions

When simplifying algebraic expressions, the goal is to make them as simple as possible.
This often involves performing operations inside parentheses first, then exponents, and then simplifying any fractions.
Let’s look at how the example problem is solved to better understand this process.
In the given problem, \((10-1)^2 - 81 \/ 2^3\), we begin by solving the inside of the parentheses: \(10-1\).
Once simplified to 9, we then square it, getting 81.
Next, we subtract 81, leading to a numerator of 0.
Simplification is all about breaking down the expression step-by-step for clarity.

Numerator and Denominator

In algebra, we often deal with fractions composed of a numerator and a denominator.
The numerator is the top part of the fraction, and the denominator is the bottom part.
In our example, after simplifying, the fraction becomes \(\frac{0}{8}\).
Here, 0 is the numerator and 8 is the denominator.
To simplify a fraction, you perform all the operations separately on both the numerator and the denominator, ensuring both are in their simplest forms.
In our case, the numerator simplified to zero, making the whole fraction equal to 0.

Order of Operations

Order of operations is crucial in algebra to solve expressions correctly.
Remember the acronym PEMDAS: Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right).
Following this order ensures that every part of the expression is dealt with systematically and accurately.
In our sample problem, \((10-1)^2 - 81 \/ 2^3\), we first handle the operation inside the parentheses: \(10-1\).
Next, we move to the exponent step, squaring the result.
We then subtract, following the simplified order, and lastly, we attend to the denominator's exponent, which gives 8.
Keeping this order prevents mistakes and leads to the correct solution.