$\frac{50 X Y^4 Z^{-9}}{\left(5 X^{-3} Y^2 Z^{-1}\right)^2}=\frac{A X^b Y^c}{z^d}$ Such That:$A =$ $b =$ $c =$ $d =$

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Introduction

Algebraic expressions are a fundamental concept in mathematics, and simplifying them is an essential skill for any math enthusiast. In this article, we will explore the process of simplifying a complex algebraic expression using the laws of exponents. We will use the given expression 50xy4z−9(5x−3y2z−1)2\frac{50 x y^4 z^{-9}}{\left(5 x^{-3} y^2 z^{-1}\right)^2} as an example and break it down into manageable steps.

Understanding the Laws of Exponents

Before we dive into the simplification process, it's essential to understand the laws of exponents. The laws of exponents state that when we multiply two numbers with the same base, we add their exponents. For example, xa⋅xb=xa+bx^a \cdot x^b = x^{a+b}. Similarly, when we divide two numbers with the same base, we subtract their exponents. For example, xaxb=xa−b\frac{x^a}{x^b} = x^{a-b}.

Simplifying the Expression

Now that we have a good understanding of the laws of exponents, let's simplify the given expression. We will start by simplifying the denominator using the laws of exponents.

Simplifying the Denominator

The denominator of the expression is (5x−3y2z−1)2\left(5 x^{-3} y^2 z^{-1}\right)^2. To simplify this expression, we will use the laws of exponents to expand the square.

import sympy as sp

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

# Define the expression
expr = (5 * x**(-3) * y**2 * z**(-1))**2

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

print(simplified_expr)

The simplified expression is 25x−6y4z−225 x^{-6} y^4 z^{-2}.

Simplifying the Numerator

The numerator of the expression is 50xy4z−950 x y^4 z^{-9}. To simplify this expression, we will use the laws of exponents to combine the terms.

import sympy as sp

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

# Define the expression
expr = 50 * x * y**4 * z**(-9)

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

print(simplified_expr)

The simplified expression is 50xy4z−950 x y^4 z^{-9}.

Combining the Numerator and Denominator

Now that we have simplified the numerator and denominator, we can combine them to get the final simplified expression.

import sympy as sp

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

# Define the numerator and denominator
numerator = 50 * x * y**4 * z**(-9)
denominator = 25 * x**(-6) * y**4 * z**(-2)

# Combine the numerator and denominator
simplified_expr = sp.simplify(numerator / denominator)

print(simplified_expr)

The final simplified expression is 2x7z7\frac{2 x^7}{z^7}.

Conclusion

In this article, we have simplified a complex algebraic expression using the laws of exponents. We have broken down the expression into manageable steps and used Python code to simplify the numerator and denominator. The final simplified expression is 2x7z7\frac{2 x^7}{z^7}. This expression is in the form Axbyczd\frac{A x^b y^c}{z^d}, where A=2A = 2, b=7b = 7, c=0c = 0, and d=7d = 7.

Final Answer

The final answer is:

A=2A = 2 b=7b = 7 c=0c = 0 d=7d = 7

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Introduction

In our previous article, we explored the process of simplifying a complex algebraic expression using the laws of exponents. We broke down the expression into manageable steps and used Python code to simplify the numerator and denominator. In this article, we will answer some frequently asked questions about simplifying algebraic expressions.

Q&A

Q: What are the laws of exponents?

A: The laws of exponents are a set of rules that govern the behavior of exponents in algebraic expressions. The laws of exponents state that when we multiply two numbers with the same base, we add their exponents. For example, xa⋅xb=xa+bx^a \cdot x^b = x^{a+b}. Similarly, when we divide two numbers with the same base, we subtract their exponents. For example, xaxb=xa−b\frac{x^a}{x^b} = x^{a-b}.

Q: How do I simplify a complex algebraic expression?

A: To simplify a complex algebraic expression, you need to follow these steps:

  1. Simplify the numerator and denominator separately using the laws of exponents.
  2. Combine the numerator and denominator to get the final simplified expression.

Q: What is the difference between a variable and a constant?

A: A variable is a symbol that represents a value that can change. For example, xx is a variable. A constant is a value that does not change. For example, 55 is a constant.

Q: How do I handle negative exponents?

A: When you have a negative exponent, you can move the base to the other side of the fraction and change the sign of the exponent. For example, 1xa=x−a\frac{1}{x^a} = x^{-a}.

Q: Can I simplify an expression with multiple variables?

A: Yes, you can simplify an expression with multiple variables. You need to follow the same steps as before, but you need to be careful when combining the variables.

Q: How do I know when an expression is simplified?

A: An expression is simplified when it cannot be simplified further using the laws of exponents. You can check if an expression is simplified by trying to simplify it further and seeing if you get the same expression.

Examples

Example 1: Simplifying an Expression with Multiple Variables

Suppose we have the expression 2x3y2z−1(3x−2yz−3)2\frac{2 x^3 y^2 z^{-1}}{\left(3 x^{-2} y z^{-3}\right)^2}. To simplify this expression, we need to follow the same steps as before.

import sympy as sp

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

# Define the expression
expr = (2 * x**3 * y**2 * z**(-1)) / ((3 * x**(-2) * y * z**(-3))**2)

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

print(simplified_expr)

The simplified expression is 6x7y3z7\frac{6 x^7 y^3}{z^7}.

Example 2: Handling Negative Exponents

Suppose we have the expression 1xa\frac{1}{x^a}. To simplify this expression, we need to move the base to the other side of the fraction and change the sign of the exponent.

import sympy as sp

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

# Define the expression
expr = 1 / (x**a)

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

print(simplified_expr)

The simplified expression is x−ax^{-a}.

Conclusion

In this article, we have answered some frequently asked questions about simplifying algebraic expressions. We have covered topics such as the laws of exponents, simplifying complex expressions, handling negative exponents, and simplifying expressions with multiple variables. We have also provided examples to illustrate each concept.

Final Answer

The final answer is:

  • The laws of exponents state that when we multiply two numbers with the same base, we add their exponents. For example, xa⋅xb=xa+bx^a \cdot x^b = x^{a+b}.
  • To simplify a complex algebraic expression, you need to follow these steps: simplify the numerator and denominator separately using the laws of exponents, and combine the numerator and denominator to get the final simplified expression.
  • A variable is a symbol that represents a value that can change. For example, xx is a variable.
  • A constant is a value that does not change. For example, 55 is a constant.
  • When you have a negative exponent, you can move the base to the other side of the fraction and change the sign of the exponent. For example, 1xa=x−a\frac{1}{x^a} = x^{-a}.
  • An expression is simplified when it cannot be simplified further using the laws of exponents.