Simplify The Expression:${ \frac{7x 7y {-3}}{35x^{-1}y} \div 5 = \frac{x 8}{5y {-2}} }$

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 process of simplifying a given expression. We will break down the problem into manageable steps, and by the end of this article, you will be able to simplify complex expressions with ease.

Understanding the Problem

The given expression is $\frac{7x^7y^{-3}}{35x^{-1}y} \div 5 = \frac{x^8}{5y^{-2}}$. Our goal is to simplify this expression by applying the rules of exponents and algebraic manipulation.

Step 1: Simplify the Coefficients

The first step in simplifying the expression is to simplify the coefficients. In this case, we have a fraction with a numerator of 7 and a denominator of 35. We can simplify the fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 7.

import sympy as sp
x = sp.symbols('x')
y = sp.symbols('y')

numerator = 7
denominator = 35
simplified_coefficient = (numerator / sp.gcd(numerator, denominator)) / (denominator / sp.gcd(numerator, denominator))
print(simplified_coefficient)

Step 2: Apply the Quotient Rule for Exponents

The next step is to apply the quotient rule for exponents, which states that when dividing two powers with the same base, we subtract the exponents. In this case, we have $x^7$ and $x^{-1}$, which can be simplified by subtracting the exponents.

# Apply the quotient rule for exponents
base = x
exponent1 = 7
exponent2 = -1
simplified_exponent = exponent1 - exponent2
print(simplified_exponent)

Step 3: Simplify the Expression

Now that we have simplified the coefficients and applied the quotient rule for exponents, we can simplify the expression by combining the results.

# Simplify the expression
simplified_expression = (simplified_coefficient * x**simplified_exponent * y**(-3)) / (5 * y)
print(simplified_expression)

Step 4: Divide the Expression by 5

The final step is to divide the expression by 5. We can do this by multiplying the expression by the reciprocal of 5.

# Divide the expression by 5
result = simplified_expression * (1/5)
print(result)

Conclusion

In this article, we simplified the given expression by applying the rules of exponents and algebraic manipulation. We broke down the problem into manageable steps and used Python code to illustrate each step. By following these steps, you can simplify complex expressions with ease and become proficient in algebraic manipulation.

Final Answer

The final answer is $\boxed{\frac{x^8}{5y^{-2}}}$.

Introduction

In our previous article, we explored the process of simplifying a given expression by applying the rules of exponents and algebraic manipulation. In this article, we will continue to delve into the world of algebra and answer some of the most frequently asked questions about simplifying expressions.

Q&A: Simplifying Expressions

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

A: The first step in simplifying an expression is to simplify the coefficients. This involves dividing both the numerator and the denominator by their greatest common divisor.

Q: How do I apply the quotient rule for exponents?

A: To apply the quotient rule for exponents, you need to subtract the exponents when dividing two powers with the same base.

Q: What is the quotient rule for exponents?

A: The quotient rule for exponents states that when dividing two powers with the same base, you subtract the exponents. For example, $x^7 \div x^3 = x^{7-3} = x^4$.

Q: How do I simplify an expression with negative exponents?

A: To simplify an expression with negative exponents, you need to apply the rule that $a^{-n} = \frac{1}{a^n}$. For example, $x^{-3} = \frac{1}{x^3}$.

Q: Can I simplify an expression with multiple variables?

A: Yes, you can simplify an expression with multiple variables by applying the rules of exponents and algebraic manipulation. For example, $\frac{x^7y^{-3}}{35x^{-1}y} \div 5 = \frac{x^8}{5y^{-2}}$.

Q: How do I divide an expression by a fraction?

A: To divide an expression by a fraction, you need to multiply the expression by the reciprocal of the fraction. For example, $\frac{x}{\frac{1}{2}} = x \times 2 = 2x$.

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

A: The final step in simplifying an expression is to combine the results and simplify the expression further if possible.

Example Questions and Answers

Q: Simplify the expression $\frac{2x^3y^2}{4x^2y^3}$.

A: To simplify the expression, we need to simplify the coefficients and apply the quotient rule for exponents.

import sympy as sp

x = sp.symbols('x')
y = sp.symbols('y')

numerator = 2
denominator = 4
simplified_coefficient = (numerator / sp.gcd(numerator, denominator)) / (denominator / sp.gcd(numerator, denominator))

base = x
exponent1 = 3
exponent2 = 2
simplified_exponent = exponent1 - exponent2

simplified_expression = (simplified_coefficient * xsimplified_exponent * y(2-3))
print(simplified_expression)

Q: Simplify the expression $\frac{x^5y^{-2}}{x^2y^3}$.

A: To simplify the expression, we need to apply the quotient rule for exponents and simplify the coefficients.

import sympy as sp

x = sp.symbols('x')
y = sp.symbols('y')

base = x
exponent1 = 5
exponent2 = 2
simplified_exponent = exponent1 - exponent2

numerator = 1
denominator = 1
simplified_coefficient = (numerator / sp.gcd(numerator, denominator)) / (denominator / sp.gcd(numerator, denominator))

simplified_expression = (simplified_coefficient * xsimplified_exponent * y(-2-3))
print(simplified_expression)

Conclusion

In this article, we answered some of the most frequently asked questions about simplifying expressions. We covered topics such as simplifying coefficients, applying the quotient rule for exponents, and simplifying expressions with negative exponents. We also provided example questions and answers to help illustrate the concepts.

Final Answer

The final answer is $\boxed{\frac{x^3y^{-5}}{1}}$.