Activity1. \[$\frac{x-2}{x-1} \times \frac{x-1}{2-x}\$\]2. \[$\frac{x-4}{2(x-4)^2} \div \frac{x-4}{4-x}\$\]3. \[$\frac{5x^2}{x+3} \div 15x\$\]4. \[$\frac{(y-1)^2}{3} \times \frac{6(x-1)}{7} \div

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Introduction

Algebraic expressions are a fundamental concept in mathematics, and solving them is a crucial skill for students and professionals alike. In this article, we will focus on solving complex algebraic expressions, specifically those involving fractions, multiplication, and division. We will break down each problem step by step, using clear and concise language to ensure that readers understand the concepts and techniques involved.

Problem 1: Multiplication of Fractions

The first problem we will tackle is the multiplication of fractions. We are given the expression:

x−2x−1×x−12−x\frac{x-2}{x-1} \times \frac{x-1}{2-x}

To solve this problem, we need to follow the order of operations (PEMDAS):

  1. Multiply the numerators: (x−2)×(x−1)=x2−3x+2(x-2) \times (x-1) = x^2 - 3x + 2
  2. Multiply the denominators: (x−1)×(2−x)=−x2+3x−2(x-1) \times (2-x) = -x^2 + 3x - 2
  3. Write the result as a fraction: x2−3x+2−x2+3x−2\frac{x^2 - 3x + 2}{-x^2 + 3x - 2}

However, we can simplify this expression further by canceling out the common factors in the numerator and denominator. In this case, we can cancel out the (x−1)(x-1) term:

x2−3x+2−x2+3x−2=(x−1)(x−2)−(x−1)(x−2)\frac{x^2 - 3x + 2}{-x^2 + 3x - 2} = \frac{(x-1)(x-2)}{-(x-1)(x-2)}

Now, we can cancel out the (x−1)(x−2)(x-1)(x-2) term:

(x−1)(x−2)−(x−1)(x−2)=1−1=−1\frac{(x-1)(x-2)}{-(x-1)(x-2)} = \frac{1}{-1} = -1

Therefore, the final answer to the first problem is −1-1.

Problem 2: Division of Fractions

The second problem we will tackle is the division of fractions. We are given the expression:

x−42(x−4)2÷x−44−x\frac{x-4}{2(x-4)^2} \div \frac{x-4}{4-x}

To solve this problem, we need to follow the order of operations (PEMDAS):

  1. Invert the second fraction: x−44−x=−(x−4)x−4=−1\frac{x-4}{4-x} = \frac{-(x-4)}{x-4} = -1
  2. Multiply the first fraction by the reciprocal of the second fraction: x−42(x−4)2×−1=x−4−2(x−4)2\frac{x-4}{2(x-4)^2} \times -1 = \frac{x-4}{-2(x-4)^2}
  3. Simplify the expression: x−4−2(x−4)2=1−2(x−4)\frac{x-4}{-2(x-4)^2} = \frac{1}{-2(x-4)}

Therefore, the final answer to the second problem is 1−2(x−4)\frac{1}{-2(x-4)}.

Problem 3: Division of a Fraction by a Polynomial

The third problem we will tackle is the division of a fraction by a polynomial. We are given the expression:

5x2x+3÷15x\frac{5x^2}{x+3} \div 15x

To solve this problem, we need to follow the order of operations (PEMDAS):

  1. Invert the second term: 15x=15x115x = \frac{15x}{1}
  2. Multiply the first fraction by the reciprocal of the second term: 5x2x+3×115x=5x215x(x+3)\frac{5x^2}{x+3} \times \frac{1}{15x} = \frac{5x^2}{15x(x+3)}
  3. Simplify the expression: 5x215x(x+3)=x3(x+3)\frac{5x^2}{15x(x+3)} = \frac{x}{3(x+3)}

Therefore, the final answer to the third problem is x3(x+3)\frac{x}{3(x+3)}.

Problem 4: Multiplication and Division of Fractions

The fourth problem we will tackle is the multiplication and division of fractions. We are given the expression:

(y−1)23×6(x−1)7÷(x−1)22\frac{(y-1)^2}{3} \times \frac{6(x-1)}{7} \div \frac{(x-1)^2}{2}

To solve this problem, we need to follow the order of operations (PEMDAS):

  1. Multiply the first two fractions: (y−1)23×6(x−1)7=(y−1)2×6(x−1)3×7=6(y−1)2(x−1)21\frac{(y-1)^2}{3} \times \frac{6(x-1)}{7} = \frac{(y-1)^2 \times 6(x-1)}{3 \times 7} = \frac{6(y-1)^2(x-1)}{21}
  2. Invert the third fraction: (x−1)22=−(x−1)2−2=−(x−1)22\frac{(x-1)^2}{2} = \frac{-(x-1)^2}{-2} = -\frac{(x-1)^2}{2}
  3. Multiply the result by the reciprocal of the third fraction: 6(y−1)2(x−1)21×−2(x−1)2=−12(y−1)221\frac{6(y-1)^2(x-1)}{21} \times -\frac{2}{(x-1)^2} = -\frac{12(y-1)^2}{21}
  4. Simplify the expression: −12(y−1)221=−4(y−1)27-\frac{12(y-1)^2}{21} = -\frac{4(y-1)^2}{7}

Therefore, the final answer to the fourth problem is −4(y−1)27-\frac{4(y-1)^2}{7}.

Conclusion

In this article, we have solved four complex algebraic expressions involving fractions, multiplication, and division. We have used the order of operations (PEMDAS) to simplify each expression and arrive at the final answer. By following these steps, readers can confidently solve similar problems and develop a deeper understanding of algebraic expressions.

Tips and Tricks

  • When solving complex algebraic expressions, it is essential to follow the order of operations (PEMDAS).
  • Cancel out common factors in the numerator and denominator to simplify the expression.
  • Invert the second term when dividing fractions.
  • Multiply the first fraction by the reciprocal of the second term when dividing fractions.
  • Simplify the expression by canceling out common factors and combining like terms.

Practice Problems

  • Solve the following expression: x2+4x+4x2−4÷x2−4x2+4x+4\frac{x^2 + 4x + 4}{x^2 - 4} \div \frac{x^2 - 4}{x^2 + 4x + 4}
  • Solve the following expression: 2x2+6x+2x2+2x+1×x2+2x+12x2+6x+2\frac{2x^2 + 6x + 2}{x^2 + 2x + 1} \times \frac{x^2 + 2x + 1}{2x^2 + 6x + 2}
  • Solve the following expression: x2−4x2+4x+4÷x2+4x+4x2−4\frac{x^2 - 4}{x^2 + 4x + 4} \div \frac{x^2 + 4x + 4}{x^2 - 4}

By practicing these problems, readers can develop their skills and confidence in solving complex algebraic expressions.

Q: What is the order of operations (PEMDAS)?

A: The order of operations (PEMDAS) is a set of rules that dictate the order in which mathematical operations should be performed when there are multiple operations in an expression. The acronym PEMDAS stands for:

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

Q: How do I simplify complex algebraic expressions?

A: To simplify complex algebraic expressions, follow these steps:

  1. Factor the numerator and denominator, if possible.
  2. Cancel out any common factors in the numerator and denominator.
  3. Simplify the expression by combining like terms.
  4. Check the expression to ensure that it is in its simplest form.

Q: What is the difference between multiplication and division of fractions?

A: When multiplying fractions, you multiply the numerators and denominators separately. When dividing fractions, you invert the second fraction and multiply.

For example:

  • Multiplication: 12×34=1×32×4=38\frac{1}{2} \times \frac{3}{4} = \frac{1 \times 3}{2 \times 4} = \frac{3}{8}
  • Division: 12÷34=12×43=46=23\frac{1}{2} \div \frac{3}{4} = \frac{1}{2} \times \frac{4}{3} = \frac{4}{6} = \frac{2}{3}

Q: How do I handle negative exponents?

A: When you have a negative exponent, you can rewrite it as a positive exponent by taking the reciprocal of the base. For example:

  • x−2=1x2x^{-2} = \frac{1}{x^2}
  • x−3=1x3x^{-3} = \frac{1}{x^3}

Q: What is the difference between a rational expression and a rational number?

A: A rational number is a number that can be expressed as the ratio of two integers, such as 34\frac{3}{4}. A rational expression, on the other hand, is an expression that can be expressed as the ratio of two polynomials, such as x+2x−1\frac{x+2}{x-1}.

Q: How do I simplify rational expressions?

A: To simplify rational expressions, follow these steps:

  1. Factor the numerator and denominator, if possible.
  2. Cancel out any common factors in the numerator and denominator.
  3. Simplify the expression by combining like terms.
  4. Check the expression to ensure that it is in its simplest form.

Q: What is the difference between a rational expression and a polynomial?

A: A polynomial is an expression that consists of one or more terms, each of which is a constant or a variable raised to a non-negative integer power. A rational expression, on the other hand, is an expression that can be expressed as the ratio of two polynomials.

For example:

  • Polynomial: x2+3x+2x^2 + 3x + 2
  • Rational expression: x+2x−1\frac{x+2}{x-1}

Q: How do I simplify rational expressions with variables?

A: To simplify rational expressions with variables, follow these steps:

  1. Factor the numerator and denominator, if possible.
  2. Cancel out any common factors in the numerator and denominator.
  3. Simplify the expression by combining like terms.
  4. Check the expression to ensure that it is in its simplest form.

Q: What is the difference between a rational expression and a complex fraction?

A: A rational expression is an expression that can be expressed as the ratio of two polynomials. A complex fraction, on the other hand, is a fraction that contains one or more fractions in its numerator or denominator.

For example:

  • Rational expression: x+2x−1\frac{x+2}{x-1}
  • Complex fraction: x+2x−1x+3\frac{\frac{x+2}{x-1}}{x+3}

Q: How do I simplify complex fractions?

A: To simplify complex fractions, follow these steps:

  1. Simplify the numerator and denominator separately.
  2. Invert the second fraction, if necessary.
  3. Multiply the numerator and denominator.
  4. Simplify the resulting expression.

Q: What is the difference between a rational expression and an algebraic expression?

A: A rational expression is an expression that can be expressed as the ratio of two polynomials. An algebraic expression, on the other hand, is a general term that refers to any expression that involves variables and constants.

For example:

  • Rational expression: x+2x−1\frac{x+2}{x-1}
  • Algebraic expression: x2+3x+2x^2 + 3x + 2

Q: How do I simplify algebraic expressions?

A: To simplify algebraic expressions, follow these steps:

  1. Combine like terms.
  2. Simplify the expression by canceling out any common factors.
  3. Check the expression to ensure that it is in its simplest form.

Q: What is the difference between a rational expression and a numerical expression?

A: A rational expression is an expression that can be expressed as the ratio of two polynomials. A numerical expression, on the other hand, is an expression that consists of only numbers.

For example:

  • Rational expression: x+2x−1\frac{x+2}{x-1}
  • Numerical expression: 3×4=123 \times 4 = 12

Q: How do I simplify numerical expressions?

A: To simplify numerical expressions, follow these steps:

  1. Perform the operations in the correct order.
  2. Simplify the expression by combining like terms.
  3. Check the expression to ensure that it is in its simplest form.

Q: What is the difference between a rational expression and a mathematical expression?

A: A rational expression is an expression that can be expressed as the ratio of two polynomials. A mathematical expression, on the other hand, is a general term that refers to any expression that involves variables, constants, and mathematical operations.

For example:

  • Rational expression: x+2x−1\frac{x+2}{x-1}
  • Mathematical expression: x2+3x+2x^2 + 3x + 2

Q: How do I simplify mathematical expressions?

A: To simplify mathematical expressions, follow these steps:

  1. Combine like terms.
  2. Simplify the expression by canceling out any common factors.
  3. Check the expression to ensure that it is in its simplest form.

Conclusion

In this article, we have answered some of the most frequently asked questions about solving complex algebraic expressions. We have covered topics such as the order of operations, simplifying rational expressions, and handling negative exponents. By following these steps and practicing with examples, you can become more confident and proficient in solving complex algebraic expressions.