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Chapter 0: Problem 50

For exercises 1-80, evaluate. $$ \frac{(14-4)^{2}}{29-2^{2}} $$

Short Answer

Expert verified
4

Step by step solution

01

Simplify the Numerator

First, simplify the expression inside the numerator: \( 14 - 4 \). This gives \( 10 \).
02

Square the Simplified Numerator

Next, square the result from the previous step: \( 10^2 \) This gives \( 100 \).
03

Simplify the Denominator

Now, simplify the expression inside the denominator: \( 29 - 2^2 \). First, calculate \( 2^2 = 4 \). Then, subtract: \( 29 - 4 = 25 \).
04

Divide the Numerator by the Denominator

Finally, divide the simplified numerator by the simplified denominator: \( \frac{100}{25} \). This gives \( 4 \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Evaluating Expressions
Evaluating expressions involves following a series of steps to simplify and find the value of the given algebraic expression. Firstly, identify all components such as numbers, variables, exponents, and operations. For instance, in the expression \( \frac{(14-4)^2}{29-2^2} \), recognize the operations nested within parentheses, exponents, numerators, and denominators.
Understanding how to handle each part is crucial. Begin with the innermost operations, often inside parentheses, to break down complex expressions. This step-by-step breakdown ensures accuracy and clarity in finding the final value.
Order of Operations
The order of operations is crucial in solving any mathematical expression correctly. The standard rule to follow is PEMDAS, which stands for:
  • Parentheses
  • Exponents
  • Multiplication and Division (from left to right)
  • Addition and Subtraction (from left to right)
Let's apply this to our expression \( \frac{(14-4)^2}{29-2^2} \).
Step 1: Parentheses
First, evaluate the expressions inside the parentheses: \(14-4 = 10 \) and \(2^2 = 4 \).
Step 2: Exponents
Then, handle the exponents: \(10^2 = 100 \).
Step 3: Multiplication and Division
Next, perform the division: \( \frac{100}{25} \).
This results in the final simplified value of \( 4 \). Following the order of operations ensures that you simplify expressions in the correct sequence, avoiding mistakes.
Simplifying Fractions
Simplifying fractions involves reducing them to their simplest form. Once you have the simplified numerator and the simplified denominator, the next step is to perform the division.
Considering our example:
  • After evaluating \( (14-4)^2 \) as \(100 \)
  • And simplifying the denominator \(29 - 2^2 = 25 \)
The fraction becomes \( \frac{100}{25} \). To simplify it:
Step 1: Check for common factors between numerator and denominator.
Step 2: Divide both the numerator and the denominator by their greatest common divisor (GCD), which is \(25 \) in this case. Hence, \( \frac{100 梅 25}{25 梅 25} = \frac{4}{1} = 4 \).
Simplifying fractions helps in better understanding and solving mathematical problems efficiently.

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Most popular questions from this chapter

For exercises \(13-24\), rewrite the fraction as a decimal number. $$ \frac{3}{8} $$

For exercises \(85-108\), write \(>\) or \(<\) between the numbers to make a true statement. $$ \frac{2}{11} \quad \frac{3}{16} $$

$$ \text { Find } 5 \% \text { of } 40 \text {. } $$

For exercises \(13-24\), rewrite the fraction as a decimal number. $$ \frac{9}{5} $$

For exercises 1-12, rewrite the decimal number as a fraction. Simplify the fraction to lowest terms. $$ 0.9 $$
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