Simplify The Expression:$\[ \frac{x^2 - 9}{4x^2} \div (x - 3) \\]

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Introduction

Algebraic manipulation is a crucial aspect of mathematics, and simplifying expressions is an essential skill that every student and mathematician should possess. In this article, we will focus on simplifying the given expression: x2−94x2÷(x−3)\frac{x^2 - 9}{4x^2} \div (x - 3). We will break down the problem into manageable steps, and by the end of this article, you will have a clear understanding of how to simplify complex algebraic expressions.

Understanding the Problem

The given expression is a division of two algebraic expressions: x2−94x2\frac{x^2 - 9}{4x^2} and (x−3)(x - 3). To simplify this expression, we need to follow the order of operations (PEMDAS):

  1. Parentheses: Evaluate expressions inside parentheses first.
  2. Exponents: Evaluate any exponential expressions next.
  3. Multiplication and Division: Evaluate multiplication and division operations from left to right.
  4. Addition and Subtraction: Finally, evaluate any addition and subtraction operations from left to right.

Step 1: Factor the Numerator

The numerator of the given expression is x2−9x^2 - 9. We can factor this expression as a difference of squares:

x2−9=(x−3)(x+3)x^2 - 9 = (x - 3)(x + 3)

Step 2: Rewrite the Expression

Now that we have factored the numerator, we can rewrite the expression as:

(x−3)(x+3)4x2÷(x−3)\frac{(x - 3)(x + 3)}{4x^2} \div (x - 3)

Step 3: Cancel Out Common Factors

We can see that the numerator and the divisor have a common factor of (x−3)(x - 3). We can cancel out this common factor:

(x+3)4x2\frac{(x + 3)}{4x^2}

Step 4: Simplify the Expression

Now that we have cancelled out the common factor, we can simplify the expression further. We can rewrite the expression as:

x+34x2\frac{x + 3}{4x^2}

Step 5: Final Simplification

The expression is now in its simplest form. We can see that there are no common factors that can be cancelled out, and the expression cannot be simplified further.

Conclusion

In this article, we have simplified the given expression: x2−94x2÷(x−3)\frac{x^2 - 9}{4x^2} \div (x - 3). We have followed the order of operations and factored the numerator to simplify the expression. By cancelling out common factors, we have arrived at the final simplified expression: x+34x2\frac{x + 3}{4x^2}. This expression cannot be simplified further, and it is now in its simplest form.

Frequently Asked Questions

  • Q: What is the order of operations? A: The order of operations is PEMDAS: Parentheses, Exponents, Multiplication and Division, and Addition and Subtraction.
  • Q: How do I factor a difference of squares? A: To factor a difference of squares, you can use the formula: a2−b2=(a−b)(a+b)a^2 - b^2 = (a - b)(a + b).
  • Q: How do I cancel out common factors? A: To cancel out common factors, you need to identify the common factors in the numerator and the divisor, and then divide both the numerator and the divisor by the common factor.

Final Thoughts

Simplifying algebraic expressions is an essential skill that every student and mathematician should possess. By following the order of operations and factoring the numerator, we can simplify complex expressions and arrive at their simplest form. In this article, we have simplified the given expression: x2−94x2÷(x−3)\frac{x^2 - 9}{4x^2} \div (x - 3). We hope that this article has provided you with a clear understanding of how to simplify complex algebraic expressions.

Introduction

In our previous article, we simplified the expression: x2−94x2÷(x−3)\frac{x^2 - 9}{4x^2} \div (x - 3). We followed the order of operations and factored the numerator to simplify the expression. In this article, we will provide a Q&A guide to algebraic manipulation, focusing on simplifying expressions.

Q&A Guide

Q: What is the order of operations?

A: The order of operations is PEMDAS: Parentheses, Exponents, Multiplication and Division, and Addition and Subtraction.

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

A: To simplify an expression with multiple operations, follow the order of operations (PEMDAS). Evaluate expressions inside parentheses first, then exponents, followed by multiplication and division, and finally addition and subtraction.

Q: How do I factor a difference of squares?

A: To factor a difference of squares, use the formula: a2−b2=(a−b)(a+b)a^2 - b^2 = (a - b)(a + b).

Q: How do I cancel out common factors?

A: To cancel out common factors, identify the common factors in the numerator and the divisor, and then divide both the numerator and the divisor by the common factor.

Q: What is the difference between simplifying and solving an equation?

A: Simplifying an equation involves reducing the equation to its simplest form, while solving an equation involves finding the value of the variable that makes the equation true.

Q: How do I simplify an expression with a fraction?

A: To simplify an expression with a fraction, follow the order of operations (PEMDAS). Evaluate expressions inside parentheses first, then exponents, followed by multiplication and division, and finally addition and subtraction.

Q: What is the rule for dividing fractions?

A: To divide fractions, invert the second fraction and multiply: ab÷cd=ab×dc\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}.

Q: How do I simplify an expression with a negative exponent?

A: To simplify an expression with a negative exponent, rewrite the expression with a positive exponent: a−n=1ana^{-n} = \frac{1}{a^n}.

Q: What is the rule for multiplying fractions?

A: To multiply fractions, multiply the numerators and denominators separately: ab×cd=acbd\frac{a}{b} \times \frac{c}{d} = \frac{ac}{bd}.

Q: How do I simplify an expression with a variable in the denominator?

A: To simplify an expression with a variable in the denominator, follow the order of operations (PEMDAS). Evaluate expressions inside parentheses first, then exponents, followed by multiplication and division, and finally addition and subtraction.

Conclusion

In this article, we have provided a Q&A guide to algebraic manipulation, focusing on simplifying expressions. We have covered topics such as the order of operations, factoring, cancelling out common factors, and simplifying expressions with fractions, negative exponents, and variables in the denominator. We hope that this article has provided you with a clear understanding of how to simplify complex algebraic expressions.

Frequently Asked Questions

  • Q: What is the difference between simplifying and solving an equation? A: Simplifying an equation involves reducing the equation to its simplest form, while solving an equation involves finding the value of the variable that makes the equation true.
  • Q: How do I simplify an expression with multiple operations? A: To simplify an expression with multiple operations, follow the order of operations (PEMDAS). Evaluate expressions inside parentheses first, then exponents, followed by multiplication and division, and finally addition and subtraction.
  • Q: What is the rule for dividing fractions? A: To divide fractions, invert the second fraction and multiply: ab÷cd=ab×dc\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}.

Final Thoughts

Simplifying algebraic expressions is an essential skill that every student and mathematician should possess. By following the order of operations and factoring the numerator, we can simplify complex expressions and arrive at their simplest form. In this article, we have provided a Q&A guide to algebraic manipulation, focusing on simplifying expressions. We hope that this article has provided you with a clear understanding of how to simplify complex algebraic expressions.