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Chapter 4: Problem 76

Divide the fractions, and simplify your result. $$\frac{18 y^{4}}{11} \div \frac{4 y^{2}}{7}$$

Short Answer

Expert verified
The result is \( \frac{63y^2}{22} \).

Step by step solution

01

Understand the Division of Fractions

To divide by a fraction, you multiply by its reciprocal. This means that \( \frac{18y^4}{11} \div \frac{4y^2}{7} \) can be rewritten as \( \frac{18y^4}{11} \times \frac{7}{4y^2} \).
02

Multiply the Numerators and Denominators

In fraction multiplication, multiply the numerators together and the denominators together: \[ \frac{18y^4 \times 7}{11 \times 4y^2} = \frac{126y^4}{44y^2}. \]
03

Simplify the Fraction

Simplify the fraction by dividing both the numerator and the denominator by the greatest common factor and canceling out common terms. In this case: \[ \frac{126y^4}{44y^2} = \frac{126 \div 2 \times y^{4-2}}{44 \div 2} = \frac{63y^2}{22}. \]
04

Final Simplification Check

Ensure that the fraction is fully simplified. In \( \frac{63y^2}{22} \), there are no common factors between 63 and 22, and the powers of \( y \) have been correctly simplified.

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Key Concepts

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

Fraction Multiplication
Fraction multiplication is a straightforward operation once you understand the key steps involved. Unlike addition or subtraction, where you need a common denominator, multiplying fractions is a bit simpler.
  • First, you multiply the numerators, which are the top numbers in each fraction.
  • Then, you multiply the denominators, which are the bottom numbers.
For example, if you have two fractions, \( \frac{a}{b} \) and \( \frac{c}{d} \), the product will be \( \frac{a \times c}{b \times d} \).
This concept is essential because it allows you to solve problems involving both fraction multiplication and division. In our exercise, multiplying \( \frac{18y^4}{11} \) by \( \frac{7}{4y^2} \) enables us to find the correct result before simplification.
Simplifying Fractions
Simplifying fractions makes them easier to interpret and use in further calculations. The aim is to make the numerator and the denominator as small as possible while maintaining the same value. This involves division by the greatest common factor (GCF) of both parts.
  • Identify the GCF of the numerator and denominator.
  • Divide both the numerator and the denominator by this number.
For example, in the fraction \( \frac{126y^4}{44y^2} \), the GCF of 126 and 44 is 2. By dividing both by this value, and applying the law of exponents \((y^4 \div y^2 = y^{4-2} = y^2)\), the simplified fraction is \( \frac{63y^2}{22} \). Remember, simplifying is crucial for clarity in your final answer.
Reciprocal of a Fraction
The concept of a reciprocal is important when dividing fractions. A reciprocal is simply switching the numerator and denominator. For any fraction \( \frac{a}{b} \), the reciprocal is \( \frac{b}{a} \).
  • Reciprocals are used to convert division problems into multiplication.
  • This makes the operations easier to manage, as multiplication is generally simpler than division for fractions.
In our example, dividing by \( \frac{4y^2}{7} \) involves finding its reciprocal \( \frac{7}{4y^2} \).
This conversion is the first step in solving fraction division problems more efficiently.

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Most popular questions from this chapter

Solve the equation and simplify your answer. $$\frac{1}{4} x-\frac{4}{3}=-\frac{2}{3}$$

Solve the equation and simplify your answer. $$\frac{2}{3} x+\frac{2}{9}=-\frac{9}{5}$$

Solve the equation and simplify your answer. $$-\frac{8}{5} x=\frac{7}{9}$$

Simplify the complex rational expression. $$\frac{\frac{5}{6}+\frac{2}{3}}{\frac{3}{5}+\frac{2}{3}}$$

At a local basketball game, announcers informed the crowd that attendance for the game was 5,394 . If this is \(2 / 7\) of the capacity, find the full seating capacity for the basketball stadium.
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