Simplify The Expression: $\[ \frac{5 X^2}{4 Y^2} \div 20 X Y \\]

Introduction

In mathematics, simplifying expressions is a crucial skill that helps us solve problems efficiently and accurately. In this article, we will focus on simplifying the given expression: $\frac{5 x^2}{4 y^2} \div 20 x y$. We will break down the steps involved in simplifying this expression and provide a clear understanding of the mathematical concepts used.

Understanding the Expression

The given expression is a division of two fractions: $\frac{5 x^2}{4 y^2}$ and $20 x y$. To simplify this expression, we need to follow the order of operations (PEMDAS):

1. Parentheses: None in this expression
2. Exponents: None in this expression
3. Multiplication and Division: Perform these operations from left to right
4. Addition and Subtraction: Perform these operations from left to right

Step 1: Rewrite the Division as a Fraction

To simplify the expression, we can rewrite the division as a fraction by inverting the second fraction and changing the division sign to multiplication:

$$\frac{5 x^2}{4 y^2} \div 20 x y = \frac{5 x^2}{4 y^2} \times \frac{1}{20 x y}$$

Step 2: Multiply the Numerators and Denominators

Now, we can multiply the numerators and denominators of the two fractions:

$$\frac{5 x^2}{4 y^2} \times \frac{1}{20 x y} = \frac{5 x^2 \times 1}{4 y^2 \times 20 x y}$$

Step 3: Simplify the Expression

We can simplify the expression by canceling out common factors in the numerator and denominator:

$$\frac{5 x^2 \times 1}{4 y^2 \times 20 x y} = \frac{5 x^2}{4 y^2 \times 20 x y}$$

Now, we can cancel out the common factor of $x$ in the numerator and denominator:

$$\frac{5 x^2}{4 y^2 \times 20 x y} = \frac{5 x}{4 y^2 \times 20 y}$$

Next, we can cancel out the common factor of $5$ in the numerator and denominator:

$$\frac{5 x}{4 y^2 \times 20 y} = \frac{x}{4 y^2 \times 4 y}$$

Finally, we can simplify the expression by canceling out the common factor of $4$ in the denominator:

$$\frac{x}{4 y^2 \times 4 y} = \frac{x}{16 y^3}$$

Conclusion

In this article, we simplified the given expression: $\frac{5 x^2}{4 y^2} \div 20 x y$. We followed the order of operations and used the concept of canceling out common factors to simplify the expression. The final simplified expression is $\frac{x}{16 y^3}$. This example demonstrates the importance of simplifying expressions in mathematics and how it can help us solve problems efficiently and accurately.

Frequently Asked Questions

Q: What is the order of operations?

A: The order of operations is a set of rules that tells us which operations to perform first when we have multiple operations in an expression. The order of operations is:

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

Q: How do I simplify an expression?

A: To simplify an expression, you can follow these steps:

1. Rewrite the expression using the order of operations
2. Cancel out common factors in the numerator and denominator
3. Simplify the expression by canceling out any remaining common factors

Q: What is the difference between a fraction and a division?

A: A fraction is a way of expressing a part of a whole, while a division is a way of expressing a quotient. In this example, we rewrote the division as a fraction by inverting the second fraction and changing the division sign to multiplication.

Additional Resources

• Khan Academy: Simplifying Expressions
• Mathway: Simplifying Expressions
• Wolfram Alpha: Simplifying Expressions

References

• "Algebra and Trigonometry" by Michael Sullivan
• "Mathematics for the Nonmathematician" by Morris Kline
• "Calculus" by Michael Spivak
  Simplify the Expression: A Step-by-Step Guide =====================================================

Q&A: Simplifying Expressions

Q: What is the order of operations?

A: The order of operations is a set of rules that tells us which operations to perform first when we have multiple operations in an expression. The order of operations is:

1. Parentheses: Evaluate expressions inside parentheses first.
2. Exponents: Evaluate any exponential expressions next (e.g., 2^3).
3. Multiplication and Division: Perform multiplication and division operations from left to right.
4. Addition and Subtraction: Finally, perform any addition and subtraction operations from left to right.

Q: How do I simplify an expression?

A: To simplify an expression, you can follow these steps:

1. Rewrite the expression: Rewrite the expression using the order of operations.
2. Cancel out common factors: Cancel out any common factors in the numerator and denominator.
3. Simplify the expression: Simplify the expression by canceling out any remaining common factors.

Q: What is the difference between a fraction and a division?

A: A fraction is a way of expressing a part of a whole, while a division is a way of expressing a quotient. In this example, we rewrote the division as a fraction by inverting the second fraction and changing the division sign to multiplication.

Q: How do I handle negative exponents?

A: When you have a negative exponent, you can rewrite it as a positive exponent by moving the base to the other side of the fraction. For example:

$$x^{-2} = \frac{1}{x^2}$$

Q: Can I simplify an expression with variables?

A: Yes, you can simplify an expression with variables. To do this, you need to follow the same steps as before:

1. Rewrite the expression using the order of operations.
2. Cancel out any common factors in the numerator and denominator.
3. Simplify the expression by canceling out any remaining common factors.

Q: How do I handle fractions with variables in the denominator?

A: When you have a fraction with variables in the denominator, you can simplify it by canceling out any common factors between the numerator and denominator. For example:

$$\frac{x^2}{x^2y} = \frac{1}{y}$$

Q: Can I simplify an expression with multiple fractions?

A: Yes, you can simplify an expression with multiple fractions. To do this, you need to follow the same steps as before:

1. Rewrite the expression using the order of operations.
2. Cancel out any common factors in the numerator and denominator.
3. Simplify the expression by canceling out any remaining common factors.

Q: How do I handle expressions with absolute values?

A: When you have an expression with absolute values, you need to consider both the positive and negative cases. For example:

$$|x| = x \text{ if } x \geq 0$$

$$|x| = -x \text{ if } x < 0$$

Q: Can I simplify an expression with radicals?

A: Yes, you can simplify an expression with radicals. To do this, you need to follow the same steps as before:

1. Rewrite the expression using the order of operations.
2. Cancel out any common factors in the numerator and denominator.
3. Simplify the expression by canceling out any remaining common factors.

Additional Resources

• Khan Academy: Simplifying Expressions
• Mathway: Simplifying Expressions
• Wolfram Alpha: Simplifying Expressions

References

• "Algebra and Trigonometry" by Michael Sullivan
• "Mathematics for the Nonmathematician" by Morris Kline
• "Calculus" by Michael Spivak