Problem 28
Question
Find the quotient. $$ \frac{2 x y z}{x^3 z^3} \div \frac{6 y^4}{2 x^2 z^2} $$
Step-by-Step Solution
Verified Answer
The simplified quotient is \( \frac{2}{3y^3} \).
1Step 1: Rewrite the Division as Multiplication
The division of fractions can be rewritten as the multiplication by the reciprocal. The reciprocal of a fraction is simply swapping the numerator and denominator. So, the given expression \( \frac{2 x y z}{x^3 z^3} \div \frac{6 y^4}{2 x^2 z^2} \) can be rewritten as \( \frac{2 x y z}{x^3 z^3} \times \frac{2 x^2 z^2}{6 y^4} \).
2Step 2: Simplify the Fractions separately
Next, simplify each fraction separately. Cancel out terms that appear in the numerator and the denominator. For the first fraction \( \frac{2 x y z}{x^3 z^3} \), cancel out x and z to get \( \frac{2 y}{x^2 z^2} \). For the second fraction \( \frac{2 x^2 z^2}{6 y^4} \), simplify 2 in numerator and 6 in denominator to get \( \frac{x^2 z^2}{3 y^4} \)
3Step 3: Multiply the Simplified Fractions
Multiply the simplified fractions from step 2. \( \frac{2 y}{x^2 z^2} \times \frac{x^2 z^2}{3 y^4} = \frac{2 y x^2 z^2}{3 y^4 x^2 z^2}\)
4Step 4: Simplify the Result
Cancel out terms that appear in both the numerator and the denominator. This gives us \( \frac{2}{3y^3} \)
Key Concepts
Reciprocal of a FractionSimplify Algebraic FractionsMultiplying Algebraic Fractions
Reciprocal of a Fraction
Understanding the concept of a reciprocal is essential when learning about fractions in algebra. The reciprocal of a fraction is created by simply switching its numerator and denominator. For instance, the reciprocal of \( \frac{a}{b} \) is \( \frac{b}{a} \) as long as none of them is zero.
When dividing fractions, you actually multiply by the reciprocal of the divisor. This is a reliable technique because division and multiplication are inverse operations. So, how does this apply to an algebraic fraction? If you have an algebraic expression like \( \frac{2xy}{x^2z^2} \) and you need its reciprocal, it would be \( \frac{x^2z^2}{2xy} \). It's important to note that any variables in the fractions follow the same rules as numbers when finding reciprocals.
When dividing fractions, you actually multiply by the reciprocal of the divisor. This is a reliable technique because division and multiplication are inverse operations. So, how does this apply to an algebraic fraction? If you have an algebraic expression like \( \frac{2xy}{x^2z^2} \) and you need its reciprocal, it would be \( \frac{x^2z^2}{2xy} \). It's important to note that any variables in the fractions follow the same rules as numbers when finding reciprocals.
Simplify Algebraic Fractions
Simplifying algebraic fractions involves reducing them to their simplest form. This process is akin to simplifying numerical fractions by canceling common factors in the numerator and denominator.
For algebraic fractions, use the properties of exponents and look for common factors that can be divided out. For example, \( \frac{x^3}{x^2} \) simplifies to \( x \) because \( x^2 \) divides out. But be careful with variables: \( \frac{x^2y}{xy^2} \) cannot be simplified to \( x \) since the \( y \) terms also affect the outcome, resulting in \( \frac{x}{y} \).
By factoring numerators and denominators, you can more easily identify and cancel out common terms. This step is crucial because it simplifies the multiplication of fractions that follows.
For algebraic fractions, use the properties of exponents and look for common factors that can be divided out. For example, \( \frac{x^3}{x^2} \) simplifies to \( x \) because \( x^2 \) divides out. But be careful with variables: \( \frac{x^2y}{xy^2} \) cannot be simplified to \( x \) since the \( y \) terms also affect the outcome, resulting in \( \frac{x}{y} \).
By factoring numerators and denominators, you can more easily identify and cancel out common terms. This step is crucial because it simplifies the multiplication of fractions that follows.
Multiplying Algebraic Fractions
Once you have simplified algebraic fractions, multiplying them is straightforward. You multiply the numerators together and the denominators together. Yet, it is prudent to look for opportunities to simplify before you actually perform the multiplication, as this can save effort and reduce the chance of errors.
Consider an example where you have the simplified fractions \( \frac{a}{b} \) and \( \frac{c}{d} \) ready to multiply. Before combining them into \( \frac{ac}{bd} \), check if \( a \) and \( d \) or \( b \) and \( c \) have common factors. If they do, reduce them. Only then should you finalize the multiplication. This approach keeps the numbers or expressions manageable and helps you reach the final, simplest form of the product.
Consider an example where you have the simplified fractions \( \frac{a}{b} \) and \( \frac{c}{d} \) ready to multiply. Before combining them into \( \frac{ac}{bd} \), check if \( a \) and \( d \) or \( b \) and \( c \) have common factors. If they do, reduce them. Only then should you finalize the multiplication. This approach keeps the numbers or expressions manageable and helps you reach the final, simplest form of the product.
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Problem 28
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