Problem 87 - Intermediate Algebra

Question

Evaluate each expression. See Example \(9 .\) $$ \frac{|-25|-2(-5)}{2^{4}-9} $$

Step-by-Step Solution

Answer

The expression evaluates to 5.

Step 1: Simplify the Numerator

The numerator of the expression is \(|-25| - 2(-5)\). The absolute value of \(-25\) is \(25\), so \(|-25| = 25\). Also, \(-2(-5)\) becomes \(10\) because multiplying two negatives gives a positive. Therefore, the numerator simplifies to \(25 + 10 = 35\).

Step 2: Simplify the Denominator

The denominator of the expression is \(2^4 - 9\). First, calculate \(2^4\), which is \(16\). Then, subtract \(9\) to get \(16 - 9 = 7\).

Step 3: Divide the Numerator by the Denominator

Now, divide the simplified numerator \(35\) by the simplified denominator \(7\). This gives \(\frac{35}{7} = 5\).

Step 4: Express the Final Result

The expression evaluates to \(5\). Therefore, the solution to the expression \(\frac{|-25|-2(-5)}{2^{4}-9}\) is \(5\).

Key Concepts

Evaluating Expressions

Understanding how to evaluate expressions is a foundational skill in algebra. It involves taking an algebraic expression and systematically working through the steps to simplify it to a final, more straightforward result.
Here, evaluating the expression means we need to simplify both the numerator and the denominator step-by-step before performing the division. Remember that within the expression, certain operations must be conducted in a specific order.

• Simplify inside any parenthesis.
• Address exponents and absolute values.
• Carry out multiplication or division from left to right.
• Finally, perform addition or subtraction from left to right.

Being careful with these steps ensures that we accurately evaluate the expression.

Absolute Value

Absolute value is an important concept that often appears in algebraic expressions. The absolute value of a number represents its distance from zero on the number line, regardless of direction.
It means that \(|-25| = 25\) because \(-25\) and \(+25\) are both 25 units away from zero.
When solving equations or evaluating expressions, absolute values transform negative numbers into positives, but leave positive numbers unchanged. This idea is crucial, especially when it comes to real-world scenarios such as measuring distances, where only the magnitude matters.

Order of Operations

The order of operations is a guideline in algebra that tells us the sequence in which we need to perform operations to correctly evaluate an expression. This sequence is often remembered by the acronym PEMDAS:

• P - Parentheses
• E - Exponents
• M/D - Multiplication and Division (from left to right)
• A/S - Addition and Subtraction (from left to right)

In simplifying the denominator \(2^4 - 9\), we first address the exponent \(2^4 = 16\), then we subtract 9 to get 7.
Proper use of order of operations avoids common mistakes such as miscalculating terms that should be addressed earlier in the process.

Algebraic Division

Algebraic division is the process of dividing one expression by another. It is often the final step in evaluating a more complex expression.
In our example, after simplifying both the numerator and the denominator, we divide \(35 \) by \(7\) to obtain \(5\). This kind of division involves breaking down each part of the expression to make the final calculation straightforward.
With algebraic division, it's essential to verify that both parts of the division are fully simplified. This ensures an accurate result and a deeper understanding of how different parts of an algebraic expression interact with each other.