Simplify The Expression: ${ \frac{(x+4)(2x-1)(x-7)}{(x+8)(2x-1)(3x-4)} \div \frac{(4x-3)(x-7)}{(x+8)(3x-4)} }$

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Introduction

Algebraic expressions are a fundamental concept in mathematics, and simplifying them is an essential skill for students and professionals alike. In this article, we will delve into the world of algebraic manipulation and provide a step-by-step guide on how to simplify the given expression. We will explore the concept of factoring, canceling out common factors, and applying the rules of exponents to arrive at the simplified expression.

Understanding the Expression

The given expression is a complex fraction, which involves the division of two rational expressions. The numerator and denominator of the first fraction are:

(x+4)(2x−1)(x−7)(x+8)(2x−1)(3x−4)\frac{(x+4)(2x-1)(x-7)}{(x+8)(2x-1)(3x-4)}

The numerator and denominator of the second fraction are:

(4x−3)(x−7)(x+8)(3x−4)\frac{(4x-3)(x-7)}{(x+8)(3x-4)}

Step 1: Factor the Numerator and Denominator

To simplify the expression, we need to factor the numerator and denominator of both fractions. Factoring involves expressing an algebraic expression as a product of simpler expressions.

import sympy as sp

x = sp.symbols('x')



numerator1 = (x+4)(2x-1)(x-7)
denominator1 = (x+8)(2x-1)(3*x-4)



numerator2 = (4x-3)(x-7)
denominator2 = (x+8)(3x-4)



factored_numerator1 = sp.factor(numerator1)
factored_denominator1 = sp.factor(denominator1)
factored_numerator2 = sp.factor(numerator2)
factored_denominator2 = sp.factor(denominator2)


print("Factored Numerator 1:", factored_numerator1)
print("Factored Denominator 1:", factored_denominator1)
print("Factored Numerator 2:", factored_numerator2)
print("Factored Denominator 2:", factored_denominator2)

Step 2: Cancel Out Common Factors

After factoring the numerator and denominator, we can cancel out common factors between the two fractions. This involves identifying the common factors in the numerator and denominator and canceling them out.

# Cancel out common factors
canceled_expression = (factored_numerator1 / factored_denominator1) / (factored_numerator2 / factored_denominator2)

print("Canceled Expression:", canceled_expression)

Step 3: Apply the Rules of Exponents

After canceling out common factors, we can apply the rules of exponents to simplify the expression further. The rules of exponents state that when we multiply two powers with the same base, we add their exponents.

# Apply the rules of exponents
simplified_expression = sp.simplify(canceled_expression)

print("Simplified Expression:", simplified_expression)

Conclusion

In this article, we have provided a step-by-step guide on how to simplify the given expression. We have explored the concept of factoring, canceling out common factors, and applying the rules of exponents to arrive at the simplified expression. By following these steps, we can simplify complex algebraic expressions and arrive at a more manageable form.

Final Answer

The final answer is (x+4)(2x−1)(4x−3)\boxed{\frac{(x+4)(2x-1)}{(4x-3)}}.

Discussion

The given expression is a complex fraction, which involves the division of two rational expressions. The numerator and denominator of the first fraction are:

(x+4)(2x−1)(x−7)(x+8)(2x−1)(3x−4)\frac{(x+4)(2x-1)(x-7)}{(x+8)(2x-1)(3x-4)}

The numerator and denominator of the second fraction are:

(4x−3)(x−7)(x+8)(3x−4)\frac{(4x-3)(x-7)}{(x+8)(3x-4)}

To simplify the expression, we need to factor the numerator and denominator of both fractions. Factoring involves expressing an algebraic expression as a product of simpler expressions.

import sympy as sp


x = sp.symbols('x')



numerator1 = (x+4)(2x-1)(x-7)
denominator1 = (x+8)(2x-1)(3*x-4)



numerator2 = (4x-3)(x-7)
denominator2 = (x+8)(3x-4)



factored_numerator1 = sp.factor(numerator1)
factored_denominator1 = sp.factor(denominator1)
factored_numerator2 = sp.factor(numerator2)
factored_denominator2 = sp.factor(denominator2)


print("Factored Numerator 1:", factored_numerator1)
print("Factored Denominator 1:", factored_denominator1)
print("Factored Numerator 2:", factored_numerator2)
print("Factored Denominator 2:", factored_denominator2)

After factoring the numerator and denominator, we can cancel out common factors between the two fractions. This involves identifying the common factors in the numerator and denominator and canceling them out.

# Cancel out common factors
canceled_expression = (factored_numerator1 / factored_denominator1) / (factored_numerator2 / factored_denominator2)

print("Canceled Expression:", canceled_expression)

After canceling out common factors, we can apply the rules of exponents to simplify the expression further. The rules of exponents state that when we multiply two powers with the same base, we add their exponents.

# Apply the rules of exponents
simplified_expression = sp.simplify(canceled_expression)

print("Simplified Expression:", simplified_expression)

The final answer is (x+4)(2x−1)(4x−3)\boxed{\frac{(x+4)(2x-1)}{(4x-3)}}.

Introduction

In our previous article, we provided a step-by-step guide on how to simplify the given expression. We explored the concept of factoring, canceling out common factors, and applying the rules of exponents to arrive at the simplified expression. In this article, we will answer some of the most frequently asked questions related to simplifying algebraic expressions.

Q&A

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

A: The first step in simplifying an algebraic expression is to factor the numerator and denominator. Factoring involves expressing an algebraic expression as a product of simpler expressions.

Q: How do I identify common factors between the numerator and denominator?

A: To identify common factors between the numerator and denominator, look for any factors that appear in both the numerator and denominator. These factors can be canceled out to simplify the expression.

Q: What are the rules of exponents?

A: The rules of exponents state that when we multiply two powers with the same base, we add their exponents. For example, if we have (x^2) * (x^3), we can simplify it to x^(2+3) = x^5.

Q: Can I simplify an algebraic expression by canceling out common factors in the numerator and denominator?

A: Yes, you can simplify an algebraic expression by canceling out common factors in the numerator and denominator. This involves identifying the common factors in the numerator and denominator and canceling them out.

Q: How do I apply the rules of exponents to simplify an algebraic expression?

A: To apply the rules of exponents to simplify an algebraic expression, look for any powers with the same base in the numerator and denominator. You can then add the exponents to simplify the expression.

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

A: The final step in simplifying an algebraic expression is to check if the expression can be simplified further. You can do this by looking for any remaining common factors or applying the rules of exponents.

Q: Can I use a calculator to simplify an algebraic expression?

A: Yes, you can use a calculator to simplify an algebraic expression. However, it's always a good idea to check your work by hand to ensure that the calculator is giving you the correct answer.

Q: How do I know if an algebraic expression is already simplified?

A: To determine if an algebraic expression is already simplified, look for any remaining common factors or powers with the same base. If you can simplify the expression further, then it is not already simplified.

Conclusion

In this article, we have answered some of the most frequently asked questions related to simplifying algebraic expressions. We have covered topics such as factoring, canceling out common factors, and applying the rules of exponents. By following these steps, you can simplify complex algebraic expressions and arrive at a more manageable form.

Final Answer

The final answer is (x+4)(2x−1)(4x−3)\boxed{\frac{(x+4)(2x-1)}{(4x-3)}}.

Discussion

Simplifying algebraic expressions is an essential skill for students and professionals alike. By following the steps outlined in this article, you can simplify complex algebraic expressions and arrive at a more manageable form. Remember to always check your work by hand to ensure that the calculator is giving you the correct answer.

Additional Resources

Final Thoughts

Simplifying algebraic expressions is a crucial skill that can be applied to a wide range of mathematical problems. By following the steps outlined in this article, you can simplify complex algebraic expressions and arrive at a more manageable form. Remember to always check your work by hand to ensure that the calculator is giving you the correct answer.