Simplify The Expression:${ \frac{y {10}}{2x 3} \cdot \frac{20x {14}}{xy 6} }$

Introduction

Algebraic manipulation is a crucial aspect of mathematics, and simplifying expressions is an essential skill that every student and professional should possess. In this article, we will delve into the world of algebra and explore the steps involved in simplifying a complex expression. We will focus on the given expression: $\frac{y^{10}}{2x^3} \cdot \frac{20x^{14}}{xy^6}$ and provide a step-by-step guide on how to simplify it.

Understanding the Expression

Before we begin simplifying the expression, let's first understand what it represents. The given expression is a product of two fractions, each with its own set of variables and exponents. The first fraction is $\frac{y^{10}}{2x^3}$, and the second fraction is $\frac{20x^{14}}{xy^6}$. Our goal is to simplify this expression by combining like terms and eliminating any unnecessary variables or exponents.

Simplifying the Expression

To simplify the expression, we need to follow a series of steps. Here's a step-by-step guide on how to simplify the given expression:

Step 1: Multiply the Numerators

The first step in simplifying the expression is to multiply the numerators of the two fractions. The numerator of the first fraction is $y^{10}$, and the numerator of the second fraction is $20x^{14}$. When we multiply these two numerators, we get:

$$y^{10} \cdot 20x^{14} = 20y^{10}x^{14}$$

Step 2: Multiply the Denominators

The next step is to multiply the denominators of the two fractions. The denominator of the first fraction is $2x^3$, and the denominator of the second fraction is $xy^6$. When we multiply these two denominators, we get:

$$2x^3 \cdot xy^6 = 2x^4y^6$$

Step 3: Combine the Numerators and Denominators

Now that we have multiplied the numerators and denominators, we can combine them to form a single fraction. The numerator of the resulting fraction is $20y^{10}x^{14}$, and the denominator is $2x^4y^6$. When we combine these two, we get:

$$\frac{20y^{10}x^{14}}{2x^4y^6}$$

Step 4: Simplify the Fraction

The final step is to simplify the fraction by eliminating any unnecessary variables or exponents. We can do this by dividing the numerator and denominator by their greatest common divisor (GCD). In this case, the GCD of $20y^{10}x^{14}$ and $2x^4y^6$ is $2x^4y^6$. When we divide both the numerator and denominator by this GCD, we get:

$$\frac{10y^4x^{10}}{1}$$

Conclusion

Simplifying the expression $\frac{y^{10}}{2x^3} \cdot \frac{20x^{14}}{xy^6}$ requires a series of steps, including multiplying the numerators and denominators, combining them to form a single fraction, and simplifying the fraction by eliminating unnecessary variables or exponents. By following these steps, we can simplify the expression and arrive at the final answer: $\frac{10y^4x^{10}}{1}$.

Tips and Tricks

Here are some tips and tricks to help you simplify expressions like the one we discussed:

• Identify like terms: When simplifying an expression, it's essential to identify like terms, which are terms that have the same variable and exponent.
• Combine like terms: Once you've identified like terms, you can combine them by adding or subtracting their coefficients.
• Eliminate unnecessary variables or exponents: When simplifying a fraction, you can eliminate unnecessary variables or exponents by dividing the numerator and denominator by their GCD.
• Use the order of operations: When simplifying an expression, it's essential to follow the order of operations (PEMDAS), which stands for Parentheses, Exponents, Multiplication and Division, and Addition and Subtraction.

Real-World Applications

Simplifying expressions like the one we discussed has numerous real-world applications. Here are a few examples:

• Physics and Engineering: Simplifying expressions is a crucial aspect of physics and engineering, where complex equations need to be solved to understand the behavior of physical systems.
• Computer Science: Simplifying expressions is also essential in computer science, where complex algorithms need to be optimized to improve performance.
• Economics: Simplifying expressions is also used in economics, where complex models need to be simplified to understand the behavior of economic systems.

Final Thoughts

Simplifying expressions like the one we discussed requires a series of steps, including multiplying the numerators and denominators, combining them to form a single fraction, and simplifying the fraction by eliminating unnecessary variables or exponents. By following these steps and using the tips and tricks we discussed, you can simplify complex expressions and arrive at the final answer. Whether you're a student or a professional, simplifying expressions is an essential skill that can help you solve complex problems and make informed decisions.

Introduction

In our previous article, we explored the steps involved in simplifying a complex expression. We discussed the importance of algebraic manipulation and provided a step-by-step guide on how to simplify the expression $\frac{y^{10}}{2x^3} \cdot \frac{20x^{14}}{xy^6}$. In this article, we will answer some of the most frequently asked questions related to simplifying expressions.

Q&A

Q: What is the first step in simplifying an expression?

A: The first step in simplifying an expression is to identify like terms, which are terms that have the same variable and exponent.

Q: How do I combine like terms?

A: To combine like terms, you need to add or subtract their coefficients. For example, if you have two terms with the same variable and exponent, you can combine them by adding or subtracting their coefficients.

Q: What is the order of operations?

A: The order of operations is a set of rules that tells you which operations to perform first when simplifying an expression. The order of operations is:

1. Parentheses
2. Exponents
3. Multiplication and Division
4. Addition and Subtraction

Q: How do I simplify a fraction?

A: To simplify a fraction, you need to divide the numerator and denominator by their greatest common divisor (GCD). This will eliminate any unnecessary variables or exponents.

Q: What is the greatest common divisor (GCD)?

A: The greatest common divisor (GCD) is the largest number that divides both the numerator and denominator of a fraction without leaving a remainder.

Q: How do I identify the GCD of two numbers?

A: To identify the GCD of two numbers, you can use the Euclidean algorithm or list the factors of each number and find the greatest common factor.

Q: Can I simplify an expression with variables and exponents?

A: Yes, you can simplify an expression with variables and exponents by following the same steps as before. You need to identify like terms, combine them, and simplify the fraction by dividing the numerator and denominator by their GCD.

Q: What are some common mistakes to avoid when simplifying expressions?

A: Some common mistakes to avoid when simplifying expressions include:

• Not identifying like terms
• Not combining like terms
• Not simplifying the fraction by dividing the numerator and denominator by their GCD
• Not following the order of operations

Q: How do I check my work when simplifying an expression?

A: To check your work when simplifying an expression, you can plug in some values for the variables and see if the expression simplifies to the expected value.

Tips and Tricks

Here are some additional tips and tricks to help you simplify expressions:

• Use a calculator: If you're having trouble simplifying an expression, try using a calculator to check your work.
• Check your work: Always check your work when simplifying an expression to make sure you haven't made any mistakes.
• Use a diagram: If you're having trouble visualizing the expression, try drawing a diagram to help you understand the relationships between the variables and exponents.
• Break down the expression: If the expression is too complex, try breaking it down into smaller parts and simplifying each part separately.

Real-World Applications

Simplifying expressions has numerous real-world applications, including:

• Physics and Engineering: Simplifying expressions is a crucial aspect of physics and engineering, where complex equations need to be solved to understand the behavior of physical systems.
• Computer Science: Simplifying expressions is also essential in computer science, where complex algorithms need to be optimized to improve performance.
• Economics: Simplifying expressions is also used in economics, where complex models need to be simplified to understand the behavior of economic systems.

Conclusion

Simplifying expressions is a crucial aspect of mathematics, and it has numerous real-world applications. By following the steps outlined in this article and using the tips and tricks provided, you can simplify complex expressions and arrive at the final answer. Whether you're a student or a professional, simplifying expressions is an essential skill that can help you solve complex problems and make informed decisions.