Practice:1. Simplify Each Product And Identify All Non-permissible Values.a) { \frac{12m^2f}{5cf} \times \frac{15c}{4m}$}$b) { \frac{3(a-b)}{(a-1)(a+5)} \times \frac{(a-5)(a+5)}{15(a-b)}$} C ) \[ C) \[ C ) \[ \frac{(y-7)(y+3)}{(2y-3)(2y+3)}

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Introduction

Simplifying complex expressions is a crucial skill in mathematics, particularly in algebra and calculus. It involves reducing complex expressions to their simplest form, making them easier to work with and understand. In this article, we will focus on simplifying three complex expressions using the rules of algebra.

Expression a: Simplifying a Fraction

The first expression we will simplify is:

12m2f5cf×15c4m\frac{12m^2f}{5cf} \times \frac{15c}{4m}

To simplify this expression, we need to follow the rules of algebra, which include multiplying like terms and canceling out common factors.

Step 1: Multiply the Numerators and Denominators

The first step in simplifying this expression is to multiply the numerators and denominators separately.

12m2f5cf×15c4m=12m2f×15c5cf×4m\frac{12m^2f}{5cf} \times \frac{15c}{4m} = \frac{12m^2f \times 15c}{5cf \times 4m}

Step 2: Cancel Out Common Factors

Now that we have multiplied the numerators and denominators, we can cancel out common factors.

12m2f×15c5cf×4m=180m2fc20cfm\frac{12m^2f \times 15c}{5cf \times 4m} = \frac{180m^2fc}{20cfm}

Step 3: Simplify the Expression

Finally, we can simplify the expression by canceling out common factors.

180m2fc20cfm=9m1\frac{180m^2fc}{20cfm} = \frac{9m}{1}

Therefore, the simplified expression is:

9m1\frac{9m}{1}

Expression b: Simplifying a Fraction with Polynomials

The second expression we will simplify is:

3(a−b)(a−1)(a+5)×(a−5)(a+5)15(a−b)\frac{3(a-b)}{(a-1)(a+5)} \times \frac{(a-5)(a+5)}{15(a-b)}

To simplify this expression, we need to follow the rules of algebra, which include multiplying like terms and canceling out common factors.

Step 1: Multiply the Numerators and Denominators

The first step in simplifying this expression is to multiply the numerators and denominators separately.

3(a−b)(a−1)(a+5)×(a−5)(a+5)15(a−b)=3(a−b)×(a−5)(a+5)(a−1)(a+5)×15(a−b)\frac{3(a-b)}{(a-1)(a+5)} \times \frac{(a-5)(a+5)}{15(a-b)} = \frac{3(a-b) \times (a-5)(a+5)}{(a-1)(a+5) \times 15(a-b)}

Step 2: Cancel Out Common Factors

Now that we have multiplied the numerators and denominators, we can cancel out common factors.

3(a−b)×(a−5)(a+5)(a−1)(a+5)×15(a−b)=3(a−5)15(a−1)\frac{3(a-b) \times (a-5)(a+5)}{(a-1)(a+5) \times 15(a-b)} = \frac{3(a-5)}{15(a-1)}

Step 3: Simplify the Expression

Finally, we can simplify the expression by canceling out common factors.

3(a−5)15(a−1)=a−55(a−1)\frac{3(a-5)}{15(a-1)} = \frac{a-5}{5(a-1)}

Therefore, the simplified expression is:

a−55(a−1)\frac{a-5}{5(a-1)}

Expression c: Simplifying a with Quadratic Expressions

The third expression we will simplify is:

(y−7)(y+3)(2y−3)(2y+3)\frac{(y-7)(y+3)}{(2y-3)(2y+3)}

To simplify this expression, we need to follow the rules of algebra, which include multiplying like terms and canceling out common factors.

Step 1: Multiply the Numerators and Denominators

The first step in simplifying this expression is to multiply the numerators and denominators separately.

(y−7)(y+3)(2y−3)(2y+3)=(y−7)(y+3)(2y−3)(2y+3)\frac{(y-7)(y+3)}{(2y-3)(2y+3)} = \frac{(y-7)(y+3)}{(2y-3)(2y+3)}

Step 2: Factor the Quadratic Expressions

Now that we have multiplied the numerators and denominators, we can factor the quadratic expressions.

(y−7)(y+3)(2y−3)(2y+3)=(y−7)(y+3)(2y−3)(2y+3)\frac{(y-7)(y+3)}{(2y-3)(2y+3)} = \frac{(y-7)(y+3)}{(2y-3)(2y+3)}

Step 3: Cancel Out Common Factors

Finally, we can cancel out common factors.

(y−7)(y+3)(2y−3)(2y+3)=y2−4y−214y2−9\frac{(y-7)(y+3)}{(2y-3)(2y+3)} = \frac{y^2-4y-21}{4y^2-9}

Therefore, the simplified expression is:

y2−4y−214y2−9\frac{y^2-4y-21}{4y^2-9}

Conclusion

Introduction

Simplifying complex expressions is a crucial skill in mathematics, particularly in algebra and calculus. In our previous article, we provided a step-by-step guide on simplifying three complex expressions using the rules of algebra. In this article, we will answer some frequently asked questions (FAQs) related to simplifying complex expressions.

Q: What are the rules of algebra for simplifying complex expressions?

A: The rules of algebra for simplifying complex expressions include:

  • Multiplying like terms
  • Canceling out common factors
  • Factoring quadratic expressions
  • Simplifying fractions

Q: How do I multiply like terms?

A: To multiply like terms, you need to multiply the coefficients (numbers in front of the variables) and add the exponents (powers of the variables). For example, if you have the expression 2x^2 + 3x^2, you can multiply the like terms by adding the coefficients and exponents: 2x^2 + 3x^2 = 5x^2.

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 denominator and cancel them out. For example, if you have the expression (x + 3) / (x + 3), you can cancel out the common factor (x + 3) by dividing both the numerator and denominator by (x + 3).

Q: How do I factor quadratic expressions?

A: To factor quadratic expressions, you need to identify the two binomials that multiply to give the quadratic expression. For example, if you have the expression x^2 + 5x + 6, you can factor it as (x + 3)(x + 2).

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

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

  • Not multiplying like terms
  • Not canceling out common factors
  • Not factoring quadratic expressions
  • Not simplifying fractions
  • Not checking for errors in the final answer

Q: How do I check my work when simplifying complex expressions?

A: To check your work when simplifying complex expressions, you need to:

  • Verify that you have multiplied like terms correctly
  • Verify that you have canceled out common factors correctly
  • Verify that you have factored quadratic expressions correctly
  • Verify that you have simplified fractions correctly
  • Verify that your final answer is correct

Q: What are some real-world applications of simplifying complex expressions?

A: Simplifying complex expressions has many real-world applications, including:

  • Physics: Simplifying complex expressions is used to solve problems in physics, such as calculating the trajectory of a projectile.
  • Engineering: Simplifying complex expressions is used to solve problems in engineering, such as designing electrical circuits.
  • Computer Science: Simplifying complex expressions is used to solve problems in computer science, such as optimizing algorithms.

Conclusion

Simplifying complex expressions is a crucial skill in mathematics, particularly in algebra and calculus. By following the rules of algebra and avoiding common mistakes, you can simplify complex expressions and make them easier to work with and understand. In this article, we have answered some frequently asked questions (FAQs) related to simplifying complex expressions.