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Magazine Chapter 4: Problem 29
Multiply the fractions, and simplify your result. $$\frac{-12 y^{3}}{13} \cdot \frac{2}{9 y^{6}}$$
Short Answer
Expert verified
The product is \( \frac{-8}{39y^3} \).
Step by step solution
01
Multiply the Numerators
To multiply the fractions \( \frac{-12 y^3}{13} \) and \( \frac{2}{9y^6} \), start by multiplying the numerators together. This gives: \(-12y^3 \times 2 = -24y^3\).
02
Multiply the Denominators
Next, multiply the denominators of the fractions: \(13 \times 9y^6 = 117y^6\).
03
Form the New Fraction
Combine the results from Step 1 and Step 2 to form a new fraction: \(\frac{-24y^3}{117y^6}\).
04
Simplify the Fraction
To simplify \( \frac{-24y^3}{117y^6} \), start by simplifying the coefficients. The greatest common divisor (GCD) of 24 and 117 is 3. Divide both the numerator and denominator by 3, resulting in \( \frac{-8y^3}{39y^6} \).
05
Simplify the Variables
Simplify \( \frac{-8y^3}{39y^6} \) by dividing both the numerator and denominator by \( y^3 \). This results in \( \frac{-8}{39y^3} \).
06
Final Simplified Result
Thus, the simplified result of the multiplication is \( \frac{-8}{39y^3} \).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Multiplying Fractions
When multiplying fractions, focus on two main parts: the numerators (the top numbers) and the denominators (the bottom numbers). Steps to Multiply Fractions:
- First, multiply the numerators of the fractions together to get the new numerator. For example, multiplying \( -12y^3 \) and \( 2 \) gives \( -24y^3 \).
- Next, multiply the denominators together to find the new denominator. In this case, \( 13 \) and \( 9y^6 \) multiply to become \( 117y^6 \).
- Finally, put the new numerator and denominator together to form a new fraction \( rac{-24y^3}{117y^6} \).
Simplifying Fractions
Simplifying fractions is the process of making a fraction as simple as possible. This means reducing it to its smallest form. Key Simplifying Steps:
- First, look at the numbers in the numerator and the denominator. Check if they have a common factor, which can be divided evenly into both.
- In our example, the GCD of \( 24 \) and \( 117 \) is \( 3 \). Dividing both by \( 3 \) simplifies \( rac{-24y^3}{117y^6} \) to \( rac{-8y^3}{39y^6} \).
- Next, look at the variables. Divide \( y^3 \) from both numerator and denominator, resulting in \( rac{-8}{39y^3} \).
Greatest Common Divisor
The greatest common divisor (GCD) is the largest number that divides two numbers without leaving a remainder. It鈥檚 crucial for simplifying fractions.Finding the GCD:
- List the factors of each number, such as \( 24 \) and \( 117 \) in this exercise.
- For \( 24 \): 1, 2, 3, 4, 6, 8, 12, and 24.
- For \( 117 \): 1, 3, 9, 13, 39, and 117.
- Identify the largest common factor. Here, it is \( 3 \).
Algebraic Fractions
Algebraic fractions contain variables as well as numbers. Working with them is similar to numeric fractions, with a few extra steps due to the presence of variables.Handling Algebraic Fractions:
- Firstly, follow multiplication rules: multiply numerators and denominators.
- Be cautious with variable terms, ensuring you combine like terms correctly. For example, \( y^3 \) multiplied by nothing more in numerator or denominator remains \( y^3 \).
- In simplification, treat variable parts like numerical parts. In \( rac{-8y^3}{39y^6} \,\ y^3 \) divides out to simplify directly to the lowest power, leaving us with \( rac{-8}{39y^3} \).
