Problem 24 - College Algebra with Modeling and Visualization

Question

Evaluate by hand. $$ \frac{13-\sqrt{9+16}}{|5-7|^{2}} $$

Step-by-Step Solution

Answer

The evaluated expression is 2.

Step 1: Evaluate the square root

First, simplify the expression inside the square root. We have \( \sqrt{9+16} \). Simplifying inside the square root gives \( 9 + 16 = 25 \). Thus, \( \sqrt{25} = 5 \).

Step 2: Substitute back the square root result

Replace \( \sqrt{9+16} \) with 5 in the original expression. The expression now becomes \( \frac{13 - 5}{|5 - 7|^2} \).

Step 3: Evaluate the numerator

Subtract 5 from 13 in the numerator: \( 13 - 5 = 8 \). So the expression is now \( \frac{8}{|5-7|^2} \).

Step 4: Evaluate the absolute value in the denominator

Calculate the absolute value of \( 5 - 7 \). First, perform the subtraction: \( 5 - 7 = -2 \). The absolute value is \( |-2| = 2 \).

Step 5: Calculate the square in the denominator

Square the absolute value obtained: \( 2^2 = 4 \). Thus, the expression becomes \( \frac{8}{4} \).

Step 6: Simplify the expression

Finally, divide 8 by 4 to simplify the expression: \( \frac{8}{4} = 2 \).

Key Concepts

Square Roots

A square root of a number is essentially a value that, when multiplied by itself, gives the original number. In this exercise, we find the square root of an expression within the parentheses. Simplifying the expression under the root, like in our example where \(9 + 16\), results in 25. The square root of 25 is 5, because \(5 \times 5 = 25\).

• Step 1: Calculate inside the square root: \(9 + 16 = 25\)
• Step 2: Find the square root: \(\sqrt{25} = 5\)

This process demonstrates how square roots work and why breaking down the expression beneath the root is crucial.

Absolute Value

The absolute value of a number refers to its distance from zero on the number line, always expressed as a non-negative. It's handy for ensuring results are positive, even if the original number inside the absolute value is negative.
In our problem, we calculate \(5 - 7\), yielding \(-2\), but its absolute value is \(2\) because distance doesn't recognize direction.

• Step 1: Perform the subtraction: \(5 - 7 = -2\)
• Step 2: Calculate the absolute value: \(|-2| = 2\)

Absolute values are vital when plugging numbers into more complex expressions because they ensure we're working with pure magnitude without regard to sign.

Order of Operations

The order of operations is a set of rules for which calculation to perform first in a multi-step math problem. It's remembered often by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division (as they appear left to right), Addition and Subtraction (as they appear left to right)).
In this exercise, adhere strictly to these rules: - First, we resolve anything in parentheses, like the operation inside the square root \((9 + 16)\).- Next comes the substitution and calculations involving exponents, such as squaring the absolute value. - Following our example:

• Resolve inside the square root, then take the root \(\sqrt{9+16}\).
• Next, evaluate the absolute value and square it \(|5-7|^2\).
• Finally, simplify the operations in any defined order as needed.

Recognizing and applying the hierarchy of steps is crucial as it helps to simplify expressions correctly and effectively.

Simplifying Expressions

Simplifying expressions involves breaking down an equation or expression to its simplest form. Each step reduces complexity, making the final expression easier to understand and solve.
In our context, it involved several steps:

• Simplify beneath any square root: \(9 + 16 = 25\).
• Evaluate and then replace parts of the expression, like turning \(\sqrt{25}\) into 5.
• Calculate and replace absolute values such as \(|5 - 7|\) with 2, then \(2^2 = 4\).
• Finally, divide 8 by 4 to get 2, which is fully simplified.

Reducing an expression not only makes math problems easier to compute but clarifies the logic behind operations. When every part is minimized, errors are less likely, ensuring accurate results.