Simplify The Expression:$\[ \frac{6a + 36}{6a} \div \frac{a^2 - 36}{a^2} \\]

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Introduction

In this article, we will simplify the given expression 6a+366a÷a2−36a2\frac{6a + 36}{6a} \div \frac{a^2 - 36}{a^2}. This involves breaking down the expression into simpler components, applying mathematical operations, and finally arriving at the simplified form. We will use algebraic manipulation and mathematical properties to simplify the expression.

Understanding the Expression

The given expression is a division of two fractions. To simplify it, we need to first understand the properties of fractions and how they can be manipulated. A fraction is a way of representing a part of a whole, and it is written in the form numeratordenominator\frac{numerator}{denominator}. When we divide one fraction by another, we can multiply the first fraction by the reciprocal of the second fraction.

Step 1: Simplify the Numerator and Denominator

The numerator of the first fraction is 6a+366a + 36, and the denominator is 6a6a. We can simplify the numerator by factoring out the common factor of 66. This gives us 6(a+6)6(a + 6). The denominator remains the same.

import sympy as sp

# Define the variable
a = sp.symbols('a')

# Simplify the numerator
numerator = 6*a + 36
simplified_numerator = sp.factor(numerator)
print(simplified_numerator)

Step 2: Simplify the Second Fraction

The numerator of the second fraction is a2−36a^2 - 36, and the denominator is a2a^2. We can simplify the numerator by factoring it as a difference of squares. This gives us (a+6)(a−6)(a + 6)(a - 6). The denominator remains the same.

# Simplify the numerator of the second fraction
numerator_2 = a**2 - 36
simplified_numerator_2 = sp.factor(numerator_2)
print(simplified_numerator_2)

Step 3: Multiply the First Fraction by the Reciprocal of the Second Fraction

Now that we have simplified the numerators and denominators, we can multiply the first fraction by the reciprocal of the second fraction. This gives us:

6(a+6)6a×a2(a+6)(a−6)\frac{6(a + 6)}{6a} \times \frac{a^2}{(a + 6)(a - 6)}

Step 4: Cancel Out Common Factors

We can cancel out the common factors between the numerator and denominator. The common factor is 6(a+6)6(a + 6), which appears in both the numerator and denominator. Canceling it out gives us:

a2a(a−6)\frac{a^2}{a(a - 6)}

Step 5: Simplify the Expression

We can simplify the expression further by canceling out the common factor of aa between the numerator and denominator. Canceling it out gives us:

aa−6\frac{a}{a - 6}

Conclusion

In this article, we simplified the given expression 6a+366a÷a2−36a2\frac{6a + 36}{6a} \div \frac{a^2 - 36}{a^2} using algebraic manipulation and mathematical properties. We broke down the expression into simpler components, applied mathematical operations, and finally arrived at the simplified form. The simplified expression is aa−6\frac{a}{a - 6}.

Final Answer

The final answer is aa−6\boxed{\frac{a}{a - 6}}.

References

Note

Introduction

In our previous article, we simplified the given expression 6a+366a÷a2−36a2\frac{6a + 36}{6a} \div \frac{a^2 - 36}{a^2} using algebraic manipulation and mathematical properties. In this article, we will answer some frequently asked questions related to the simplification of the expression.

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

A: The first step in simplifying the expression is to simplify the numerator and denominator of the first fraction. We can simplify the numerator by factoring out the common factor of 66. This gives us 6(a+6)6(a + 6). The denominator remains the same.

Q: How do we simplify the second fraction?

A: We can simplify the numerator of the second fraction by factoring it as a difference of squares. This gives us (a+6)(a−6)(a + 6)(a - 6). The denominator remains the same.

Q: What is the next step in simplifying the expression?

A: The next step in simplifying the expression is to multiply the first fraction by the reciprocal of the second fraction. This gives us:

6(a+6)6a×a2(a+6)(a−6)\frac{6(a + 6)}{6a} \times \frac{a^2}{(a + 6)(a - 6)}

Q: How do we cancel out common factors?

A: We can cancel out the common factors between the numerator and denominator. The common factor is 6(a+6)6(a + 6), which appears in both the numerator and denominator. Canceling it out gives us:

a2a(a−6)\frac{a^2}{a(a - 6)}

Q: What is the final simplified expression?

A: The final simplified expression is aa−6\frac{a}{a - 6}.

Q: Why is it important to simplify expressions?

A: Simplifying expressions is important because it helps us to:

  • Reduce the complexity of the expression
  • Make it easier to understand and work with
  • Avoid errors and mistakes
  • Improve the accuracy of calculations

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

A: An expression is simplified when it cannot be reduced further. This means that there are no common factors that can be canceled out, and the expression is in its simplest form.

Q: Can I use a calculator to simplify expressions?

A: Yes, you can use a calculator to simplify expressions. However, it's always a good idea to double-check your work and make sure that the expression is simplified correctly.

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

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

  • Not canceling out common factors
  • Not simplifying the numerator and denominator
  • Not using the correct mathematical operations
  • Not checking the work for errors

Conclusion

In this article, we answered some frequently asked questions related to the simplification of the expression 6a+366a÷a2−36a2\frac{6a + 36}{6a} \div \frac{a^2 - 36}{a^2}. We covered topics such as simplifying the numerator and denominator, canceling out common factors, and the importance of simplifying expressions. We also provided some tips and tricks for avoiding common mistakes when simplifying expressions.

Final Answer

The final answer is aa−6\boxed{\frac{a}{a - 6}}.

References

Note

This article is for educational purposes only and is not intended to be used as a substitute for professional mathematical advice. If you have any questions or need further clarification, please feel free to ask.