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Chapter 4: Problem 619

In the following exercises, simplify. $$\frac{8 \cdot 4-5^{2}}{3 \cdot 12}$$

Short Answer

Expert verified
The simplified form is \(\frac{7}{36}\).

Step by step solution

01

Title - Simplify the Numerator

First, simplify the expression in the numerator. The numerator is given as \(8 \times 4 - 5^2\). Calculate each part separately: \(8 \times 4 = 32\) and \(5^2 = 25\). Then, subtract the results: \(32 - 25 = 7\). So the simplified numerator is 7.
02

Title - Simplify the Denominator

Next, simplify the expression in the denominator. The denominator is given as \(3 \times 12\). Calculate the multiplication: \(3 \times 12 = 36\). So the simplified denominator is 36.
03

Title - Form the Simplified Fraction

Now, form the simplified fraction using the simplified numerator and denominator. This gives us \(\frac{7}{36}\).

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Key Concepts

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

Prealgebra
Prealgebra forms the foundation for all future math courses. It covers basic arithmetic involving integers, fractions, and decimals. One critical part of prealgebra is understanding how to simplify expressions and fractions.
When given an expression like \[ \frac{8 \times 4 - 5^2}{3 \times 12} \], prealgebra skills help us tackle it step-by-step. This involves breaking down the arithmetic operations and simplifying both the numerator and the denominator.
Mastering prealgebra concepts ensures you are well-prepared for more advanced math topics, including algebra and calculus.
Numerator and Denominator
A fraction consists of two parts: the numerator and the denominator. The numerator is the top part of the fraction, and it indicates how many parts of the whole are taken. The denominator is the bottom part, showing the total number of equal parts the whole is divided into.
For instance, in the fraction \[ \frac{8 \times 4 - 5^2}{3 \times 12} \], we need to simplify both the numerator and the denominator separately.
  • The numerator is simplified to 7 by calculating \(8 \times 4 = 32 \) and \(5^2 = 25\), then finding \(32 - 25 = 7\).
  • The denominator is simplified to 36 by calculating \(3 \times 12\).
So, the simplified fraction is \[ \frac{7}{36} \]. Understanding these parts helps in performing the correct operations when solving math problems.
Step-by-Step Problem Solving
Problem-solving in math often involves breaking down complex expressions into simpler steps. For our exercise, simplifying \[ \frac{8 \times 4 - 5^2}{3 \times 12}\], is achieved step by step as follows:
  • Step 1: Simplify the numerator. Calculate \(8 \times 4 = 32\), then \(5^2 = 25\). Finally, subtract to get the simplified numerator: \(32 - 25 = 7\).
  • Step 2: Simplify the denominator. Calculate \(3 \times 12 = 36\).
  • Step 3: Form the final fraction. Using the simplified numerator (7) and the denominator (36), we get \[ \frac{7}{36} \].
This approach ensures that each part of the problem is addressed in an organized manner, making difficult tasks easier to manage. Always work systematically and check your calculations for accuracy.

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Most popular questions from this chapter

In the following exercises, add. $$\frac{15}{32}+\frac{9}{32}$$

In the following exercises, evaluate the given expression. Express your answers in simplified form, using improper fractions if necessary. \(x-\frac{1}{3}\) when (a) \(x=\frac{2}{3} \quad\) (b) \(x=-\frac{2}{3}\)

In the following exercises, solve the equation. $$ x-\frac{1}{6}=\frac{7}{6} $$

In the following exercises, evaluate the given expression. Express your answers in simplified form, using improper fractions if necessary. \(\frac{u+v}{w}\) when \(u=-4, v=-8, w=2\)

In the following exercises, translate and solve. Three-eighths of \(y\) is 24
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