Simplify The Expression:$\frac{27 X^2 Y^4}{16 Y Z^3} \cdot \frac{8 Z}{9 X Y^3}$

Introduction

In this article, we will simplify the given expression $\frac{27 x^2 y^4}{16 y z^3} \cdot \frac{8 z}{9 x y^3}$. This involves applying the rules of exponents and simplifying the resulting expression. We will break down the process into manageable steps, making it easier to understand and follow along.

Step 1: Multiply the Numerators and Denominators

To simplify the given expression, we need to multiply the numerators and denominators separately. The numerator of the first fraction is $27 x^2 y^4$, and the numerator of the second fraction is $8 z$. The denominator of the first fraction is $16 y z^3$, and the denominator of the second fraction is $9 x y^3$.

import sympy as sp

# Define the variables
x, y, z = sp.symbols('x y z')

# Define the expression
expr = (27*x**2*y**4)/(16*y*z**3) * (8*z)/(9*x*y**3)

# Simplify the expression
simplified_expr = sp.simplify(expr)

Step 2: Apply the Rules of Exponents

When multiplying fractions, we can add the exponents of the same base. We will apply this rule to simplify the expression.

# Apply the rules of exponents
simplified_expr = simplified_expr.simplify()

Step 3: Cancel Out Common Factors

After applying the rules of exponents, we can cancel out any common factors between the numerator and denominator.

# Cancel out common factors
simplified_expr = simplified_expr.cancel()

Step 4: Simplify the Expression

Now that we have applied the rules of exponents and canceled out common factors, we can simplify the expression further.

# Simplify the expression
simplified_expr = simplified_expr.simplify()

Conclusion

In this article, we simplified the given expression $\frac{27 x^2 y^4}{16 y z^3} \cdot \frac{8 z}{9 x y^3}$ using the rules of exponents and canceling out common factors. We broke down the process into manageable steps, making it easier to understand and follow along.

Final Answer

The final simplified expression is $\boxed{\frac{2x^2y}{3z^2}}$.

Example Use Cases

This simplified expression can be used in various mathematical contexts, such as:

• Calculating the area of a rectangle with dimensions $x$ and $y$
• Finding the volume of a cylinder with radius $z$ and height $x$
• Determining the surface area of a sphere with radius $z$

Tips and Variations

• When simplifying expressions, it's essential to apply the rules of exponents and cancel out common factors to ensure accuracy.
• The order of operations (PEMDAS) should be followed when simplifying expressions.
• This technique can be applied to more complex expressions involving multiple variables and operations.

Common Mistakes

• Failing to apply the rules of exponents when simplifying expressions.
• Not canceling out common factors between the numerator and denominator.
• Not following the order of operations (PEMDAS) when simplifying expressions.

Further Reading

For more information on simplifying expressions and applying the rules of exponents, refer to the following resources:

• Khan Academy: Exponents and Exponential Functions
• Mathway: Simplifying Expressions
• Wolfram Alpha: Simplifying Expressions

Introduction

In our previous article, we simplified the expression $\frac{27 x^2 y^4}{16 y z^3} \cdot \frac{8 z}{9 x y^3}$ using the rules of exponents and canceling out common factors. In this article, we will answer some frequently asked questions related to simplifying expressions and provide additional tips and resources.

Q: What are the rules of exponents?

A: The rules of exponents are a set of mathematical rules that govern the behavior of exponents when they are multiplied or divided. The main rules are:

• Product of Powers Rule: When multiplying two powers with the same base, add the exponents.
• Power of a Power Rule: When raising a power to a power, multiply the exponents.
• Quotient of Powers Rule: When dividing two powers with the same base, subtract the exponents.

Q: How do I simplify an expression with multiple variables?

A: To simplify an expression with multiple variables, follow these steps:

1. Apply the rules of exponents: Use the product of powers rule, power of a power rule, and quotient of powers rule to simplify the expression.
2. Cancel out common factors: Cancel out any common factors between the numerator and denominator.
3. Simplify the expression: Use the simplified expression to calculate the final answer.

Q: What is the order of operations (PEMDAS)?

A: The order of operations (PEMDAS) is a set of rules that govern the order in which mathematical operations should be performed. The acronym PEMDAS stands for:

• P: Parentheses
• E: Exponents
• M: Multiplication
• D: Division
• A: Addition
• S: Subtraction

Q: How do I use the order of operations to simplify an expression?

A: To use the order of operations to simplify an expression, follow these steps:

1. Evaluate expressions inside parentheses: Evaluate any expressions inside parentheses first.
2. Evaluate exponents: Evaluate any exponents next.
3. Perform multiplication and division: Perform any multiplication and division operations from left to right.
4. Perform addition and subtraction: Finally, perform any addition and subtraction operations from left to right.

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

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

• Failing to apply the rules of exponents: Make sure to apply the product of powers rule, power of a power rule, and quotient of powers rule when simplifying expressions.
• Not canceling out common factors: Make sure to cancel out any common factors between the numerator and denominator.
• Not following the order of operations (PEMDAS): Make sure to follow the order of operations (PEMDAS) when simplifying expressions.

Q: Where can I find additional resources for simplifying expressions?

A: Some additional resources for simplifying expressions include:

• Khan Academy: Exponents and Exponential Functions
• Mathway: Simplifying Expressions
• Wolfram Alpha: Simplifying Expressions
• Math textbooks: Many math textbooks include chapters on simplifying expressions and applying the rules of exponents.

Conclusion

In this article, we answered some frequently asked questions related to simplifying expressions and provided additional tips and resources. By following the rules of exponents and canceling out common factors, you can simplify complex expressions and arrive at the final answer with confidence.