Problem 51 - Intermediate Algebra

Question

Divide, and then simplify, if possible. See Example 6. $$ \frac{22 x^{3}}{y^{2}} \div \frac{33 x^{9}}{y^{7}} $$

Step-by-Step Solution

Answer

The simplified result is \( \frac{2y^5}{3x^6} \).

Step 1: Understanding Division of Fractions

To divide by a fraction, you multiply by its reciprocal. Therefore, divide \( \frac{22x^3}{y^2} \) by \( \frac{33x^9}{y^7} \) means multiplying \( \frac{22x^3}{y^2} \) by the reciprocal of \( \frac{33x^9}{y^7} \), which is \( \frac{y^7}{33x^9} \).

Step 2: Writing as Multiplication

Replace the division with multiplication by the reciprocal: \[ \frac{22x^3}{y^2} \times \frac{y^7}{33x^9}. \]

Step 3: Multiply the Fractions

When multiplying fractions, multiply the numerators together and the denominators together. This gives: \[ \frac{22x^3 \cdot y^7}{y^2 \cdot 33x^9}. \]

Step 4: Simplify the Expression

First, simplify the coefficients: \( \frac{22}{33} = \frac{2}{3} \). Next, simplify the variables. For \( x \), use the property \( x^a \div x^b = x^{a-b} \): - \( x^3 \div x^9 = x^{3-9} = x^{-6} = \frac{1}{x^6} \).For \( y \), - \( y^7 \div y^2 = y^{7-2} = y^5 \). So the simplified expression is: \[ \frac{2y^5}{3x^6}. \]

Key Concepts

Fraction Division

Fraction division might sound tricky, but it's quite approachable when you break it down into parts. The key idea is to change the task from division to multiplication. The rule is simple: dividing by a fraction is the same as multiplying by its reciprocal.

This means that if you have an expression like \( \frac{22 x^{3}}{y^{2}} \div \frac{33 x^{9}}{y^{7}} \), instead of dividing, you flip the second fraction and multiply. This makes finding the solution much easier to manage.

• The operation of division gets transformed into a multiplication.
• You work with the reciprocal of the fraction you are dividing by.
• Once transformed, you apply standard multiplication rules to find the result.

By thinking of fraction division in terms of multiplication, you can streamline your approach and eliminate unnecessary confusion.

Reciprocal

The reciprocal is fundamental when thinking about fraction division. A reciprocal is what you get when you switch a fraction's numerator and denominator.

For example, the reciprocal of \( \frac{33 x^{9}}{y^{7}} \) is \( \frac{y^{7}}{33 x^{9}} \). When you find the reciprocal, you make division problems far more practical, transforming them into multiplication operations.

• To find the reciprocal of a fraction, swap its top and bottom numbers (numerator and denominator).
• This technique allows you to conduct division by multiplication.

Using a reciprocal helps simplify your workflow, transforming an intimidating division into something familiar and easy to handle.

Simplifying Expressions

Once you have transformed a division problem into a multiplication problem using reciprocals, you should simplify the expression. This makes the numbers smaller and the computation easier.

Simplifying involves reducing fractions by removing any common factors from the numerator and denominator. In terms of our example, multiplying the fractions looks like: \[ \frac{22x^3 \cdot y^7}{y^2 \cdot 33x^9}. \] You can then simplify it further:

• Check the coefficients: \( \frac{22}{33} \) simplifies to \( \frac{2}{3} \).
• Simplify variable terms by using exponent rules, which are explained right next.

These steps give a clearer, easier expression to work with, making algebraic operations smoother.

Exponent Rules

Exponent rules are crucial for simplifying expressions, particularly with variables. They dictate how you handle powers during operations.

One essential rule here involves division: - When you divide like bases, subtract their exponents: \[ x^a \div x^b = x^{a-b} \]. In our example, applying these rules:

• For \(x\): \(x^3 \div x^9 = x^{3-9} = x^{-6}\), which can be written as \(\frac{1}{x^6}\).
• For \(y\): \(y^7 \div y^2 = y^{7-2} = y^5\).

Using these rules efficiently simplifies expressions drastically, reducing complex-looking terms to straightforward outcomes. Applying exponent rules is like a shortcut in math, getting you from a complicated expression to a simplified solution effortlessly.