Simplify The Expression: $ \left(\frac{-4 Y X^2}{16 X Y 3}\right) 2 $

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Understanding the Problem

When simplifying an expression, it's essential to start by analyzing the given expression and identifying the operations involved. In this case, we are given the expression $\left(\frac{-4 y x^2}{16 x y^3}\right)^2$. Our goal is to simplify this expression, which involves squaring the fraction.

Breaking Down the Expression

To simplify the expression, we need to break it down into smaller components. The given expression can be rewritten as $\left(\frac{-4}{16}\right)^2 \cdot \left(\frac{y}{y^3}\right)^2 \cdot \left(\frac{x^2}{x}\right)^2$. This allows us to separate the expression into three distinct parts: the coefficient, the variable part, and the power of x.

Simplifying the Coefficient

The coefficient part of the expression is $\left(\frac{-4}{16}\right)^2$. To simplify this, we can start by simplifying the fraction $\frac{-4}{16}$. We can do this by dividing both the numerator and the denominator by their greatest common divisor, which is 4. This gives us $\frac{-1}{4}$. Now, we can square this fraction: $\left(\frac{-1}{4}\right)^2 = \frac{1}{16}$.

Simplifying the Variable Part

The variable part of the expression is $\left(\frac{y}{y^3}\right)^2$. To simplify this, we can start by simplifying the fraction $\frac{y}{y^3}$. We can do this by dividing both the numerator and the denominator by their greatest common divisor, which is y. This gives us $\frac{1}{y^2}$. Now, we can square this fraction: $\left(\frac{1}{y^2}\right)^2 = \frac{1}{y^4}$.

Simplifying the Power of x

The power of x part of the expression is $\left(\frac{x^2}{x}\right)^2$. To simplify this, we can start by simplifying the fraction $\frac{x^2}{x}$. We can do this by dividing both the numerator and the denominator by their greatest common divisor, which is x. This gives us $x$. Now, we can square this: $x^2$.

Combining the Simplified Parts

Now that we have simplified each part of the expression, we can combine them to get the final simplified expression. We have $\frac{1}{16} \cdot \frac{1}{y^4} \cdot x^2$. To combine these fractions, we can multiply the numerators and the denominators: $\frac{1 \cdot 1 \cdot x^2}{16 \cdot y^4} = \frac{x^2}{16y^4}$.

Conclusion

In conclusion, the simplified expression is $\frac{x^2}{16y^4}$. This expression is the result of simplifying the original expression $\left(\frac{-4 y x^2}{16 x y^3}\right)^2$.

Final Answer

The final answer is $\boxed{\frac{x^2}{16y^4}}$.

Step-by-Step Solution

Here's a step-by-step solution to the problem:

1. Break down the expression into smaller components.
2. Simplify the coefficient part of the expression.
3. Simplify the variable part of the expression.
4. Simplify the power of x part of the expression.
5. Combine the simplified parts to get the final simplified expression.

Common Mistakes

When simplifying expressions, it's essential to avoid common mistakes such as:

• Not simplifying the coefficient part of the expression.
• Not simplifying the variable part of the expression.
• Not simplifying the power of x part of the expression.
• Not combining the simplified parts correctly.

Tips and Tricks

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

• Start by breaking down the expression into smaller components.
• Simplify each part of the expression separately.
• Use the order of operations (PEMDAS) to simplify the expression.
• Combine the simplified parts to get the final simplified expression.

Real-World Applications

Simplifying expressions has many real-world applications, such as:

• Simplifying algebraic expressions in mathematics.
• Simplifying complex fractions in finance.
• Simplifying expressions in physics and engineering.

Conclusion

In conclusion, simplifying expressions is an essential skill in mathematics and has many real-world applications. By following the steps outlined in this article, you can simplify expressions and solve problems with ease.

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Frequently Asked Questions

Q: What is the first step in simplifying the expression $\left(\frac{-4 y x^2}{16 x y^3}\right)^2$?

A: The first step in simplifying the expression is to break it down into smaller components. This involves separating the expression into the coefficient, the variable part, and the power of x.

Q: How do I simplify the coefficient part of the expression?

A: To simplify the coefficient part of the expression, you need to simplify the fraction $\frac{-4}{16}$. You can do this by dividing both the numerator and the denominator by their greatest common divisor, which is 4. This gives you $\frac{-1}{4}$. Now, you can square this fraction: $\left(\frac{-1}{4}\right)^2 = \frac{1}{16}$.

Q: How do I simplify the variable part of the expression?

A: To simplify the variable part of the expression, you need to simplify the fraction $\frac{y}{y^3}$. You can do this by dividing both the numerator and the denominator by their greatest common divisor, which is y. This gives you $\frac{1}{y^2}$. Now, you can square this fraction: $\left(\frac{1}{y^2}\right)^2 = \frac{1}{y^4}$.

Q: How do I simplify the power of x part of the expression?

A: To simplify the power of x part of the expression, you need to simplify the fraction $\frac{x^2}{x}$. You can do this by dividing both the numerator and the denominator by their greatest common divisor, which is x. This gives you $x$. Now, you can square this: $x^2$.

Q: How do I combine the simplified parts to get the final simplified expression?

A: To combine the simplified parts, you need to multiply the numerators and the denominators: $\frac{1 \cdot 1 \cdot x^2}{16 \cdot y^4} = \frac{x^2}{16y^4}$.

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

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

• Not simplifying the coefficient part of the expression.
• Not simplifying the variable part of the expression.
• Not simplifying the power of x part of the expression.
• Not combining the simplified parts correctly.

Q: What are some tips and tricks to help me simplify expressions?

A: Some tips and tricks to help you simplify expressions include:

• Start by breaking down the expression into smaller components.
• Simplify each part of the expression separately.
• Use the order of operations (PEMDAS) to simplify the expression.
• Combine the simplified parts to get the final simplified expression.

Q: What are some real-world applications of simplifying expressions?

A: Simplifying expressions has many real-world applications, such as:

• Simplifying algebraic expressions in mathematics.
• Simplifying complex fractions in finance.
• Simplifying expressions in physics and engineering.

Additional Resources

If you're struggling to simplify expressions, here are some additional resources that may help:

• Online algebra calculators
• Math textbooks and workbooks
• Online math courses and tutorials
• Math apps and software

Conclusion

In conclusion, simplifying expressions is an essential skill in mathematics and has many real-world applications. By following the steps outlined in this article and avoiding common mistakes, you can simplify expressions and solve problems with ease.