Problem 44 - Prealgebra

Question

Simplify each expression as much as possible. $$18 \div\left(\frac{3}{4}\right)^{2}+49 \div\left(\frac{7}{9}\right)^{2}$$

Step-by-Step Solution

Answer

The simplified expression is 113.

Step 1: Simplify the First Division

First, we need to simplify the expression \( 18 \div \left( \frac{3}{4} \right)^2 \). Start by calculating \( \left( \frac{3}{4} \right)^2 \), which is \( \frac{3}{4} \times \frac{3}{4} = \frac{9}{16} \). Next, perform the division \( 18 \div \frac{9}{16} \), by multiplying 18 by the reciprocal of \( \frac{9}{16} \), giving us \( 18 \times \frac{16}{9} \). This simplifies to \( 32 \).

Step 2: Simplify the Second Division

Now simplify the expression \( 49 \div \left( \frac{7}{9} \right)^2 \). Calculate \( \left( \frac{7}{9} \right)^2 \), which is \( \frac{7}{9} \times \frac{7}{9} = \frac{49}{81} \). Then, calculate \( 49 \div \frac{49}{81} \) by multiplying 49 by the reciprocal \( \frac{81}{49} \), resulting in \( 49 \times \frac{81}{49} \). This simplifies to \( 81 \).

Step 3: Add the Results

Now that both divisions are simplified, add the results from Step 1 and Step 2. The expression becomes \( 32 + 81 \), which sums to \( 113 \).

Key Concepts

Arithmetic Operations

Arithmetic operations form the backbone of mathematics and are essential for solving expressions in prealgebra. These operations include addition, subtraction, multiplication, and division. Each operation helps us transform and simplify mathematical expressions. Understanding these operations and how they interact is crucial for tackling more complex problems.

• Addition involves combining two or more numbers to get a total value.
• Subtraction is about finding the difference between numbers.
• Multiplication is repeated addition and simplifies how we convey quantities of the same thing.
• Division involves splitting a number into equal parts or groups.

By mastering arithmetic operations, we lay the foundation for handling more advanced concepts like those covered in prealgebra exercises.

Simplifying Expressions

Simplifying expressions means making a mathematical expression as simple as possible. This involves combining like terms, reducing fractions, and performing arithmetic operations. The goal is to make the expression easier to work with and understand. In the problem at hand, simplifying involves calculating powers and performing divisions effectively.
Simplifying often requires several steps:

• Identify what needs simplifying, such as terms enclosed in parentheses.
• Apply arithmetic operations to these terms to reduce them.
• Combine similar terms to get the simplest form of the expression.

This step-by-step approach helps tackle complex problems by breaking them down into more manageable parts.

Fractions and Division

Fractions represent a part of a whole and are an essential part of division in mathematics. Division by fractions is one of the core skills in prealgebra, which sometimes requires the use of a reciprocal. When dividing by a fraction, like in our exercise, it's crucial to understand how fractions work. A fraction

• has a numerator (top part).
• and a denominator (bottom part).

When dividing by a fraction, you multiply by its reciprocal. This might simplify your problems greatly, such as turning a division operation into an easier multiplication because "dividing by a fraction is the same as multiplying by its reciprocal." Understanding this principle will help in efficiently solving similar exercises regularly.

Reciprocal

A reciprocal is derived by flipping the numerator and the denominator of a fraction. It is a fundamental concept in division involving fractions. The reciprocal is used to turn a division problem into a multiplication problem, simplifying the process considerably.For example, the reciprocal of the fraction \( \frac{9}{16} \) is \( \frac{16}{9} \). This reciprocal transformation is key in simplifying division within expressions.

• To find the reciprocal of a fraction, swap its numerator and denominator.
• For a whole number \( a \), its reciprocal is \( \frac{1}{a} \).

Utilizing the reciprocal makes dealing with complex fractional division intuitive and less cumbersome in calculations.

Multiplication of Fractions

Multiplying fractions involves multiplying the numerators together and the denominators together. This straightforward process forms the basis of simplifying expressions with fractions.

• Multiply numerators: If you have \( \frac{a}{b} \) and \( \frac{c}{d} \), multiply \( a \times c \).
• Multiply denominators: Multiply \( b \times d \) to find the new denominator.
• Always simplify the result when possible, dividing both the numerator and the denominator by their greatest common divisor.

When you master multiplication of fractions, dealing with problems such as the one provided becomes intuitive and less error-prone.