Select The Correct Answer.What Is This Expression In Simplest Form? 4 V W − 2 + 3 W W − 3 \frac{4v}{w-2} + \frac{3w}{w-3} W − 2 4 V ​ + W − 3 3 W ​ A. 7 W W 2 + 6 \frac{7w}{w^2+6} W 2 + 6 7 W ​ B. 7 W 2 − 18 W W 2 − 5 W + 6 \frac{7w^2-18w}{w^2-5w+6} W 2 − 5 W + 6 7 W 2 − 18 W ​ C. 7 W 2 − 5 W 2 − 5 W + 6 \frac{7w^2-5}{w^2-5w+6} W 2 − 5 W + 6 7 W 2 − 5 ​ D. 7 W 2 W − 5 \frac{7w}{2w-5} 2 W − 5 7 W ​

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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 focus on simplifying a specific expression, 4vw−2+3ww−3\frac{4v}{w-2} + \frac{3w}{w-3}, and explore the different options provided.

Understanding the Expression

The given expression is a sum of two fractions, each with a different denominator. To simplify this expression, we need to find a common denominator and then combine the fractions.

Step 1: Find a Common Denominator

To find a common denominator, we need to identify the least common multiple (LCM) of the two denominators, w−2w-2 and w−3w-3. The LCM of these two expressions is (w−2)(w−3)(w-2)(w-3).

Step 2: Rewrite the Fractions

Now that we have the common denominator, we can rewrite each fraction with the new denominator.

4vw−2=4v(w−3)(w−2)(w−3)\frac{4v}{w-2} = \frac{4v(w-3)}{(w-2)(w-3)}

3ww−3=3w(w−2)(w−2)(w−3)\frac{3w}{w-3} = \frac{3w(w-2)}{(w-2)(w-3)}

Step 3: Combine the Fractions

Now that we have rewritten each fraction with the common denominator, we can combine them.

4v(w−3)(w−2)(w−3)+3w(w−2)(w−2)(w−3)=4v(w−3)+3w(w−2)(w−2)(w−3)\frac{4v(w-3)}{(w-2)(w-3)} + \frac{3w(w-2)}{(w-2)(w-3)} = \frac{4v(w-3) + 3w(w-2)}{(w-2)(w-3)}

Simplifying the Numerator

To simplify the numerator, we can expand and combine like terms.

4v(w−3)+3w(w−2)=4vw−12v+3w2−6w4v(w-3) + 3w(w-2) = 4vw - 12v + 3w^2 - 6w

Simplifying the Expression

Now that we have simplified the numerator, we can rewrite the expression.

4vw−12v+3w2−6w(w−2)(w−3)\frac{4vw - 12v + 3w^2 - 6w}{(w-2)(w-3)}

Factoring the Numerator

To simplify the expression further, we can factor the numerator.

4vw−12v+3w2−6w=(4v−6)(w−1)+(3w−6)(w−1)4vw - 12v + 3w^2 - 6w = (4v - 6)(w - 1) + (3w - 6)(w - 1)

=(4v−6+3w−6)(w−1)= (4v - 6 + 3w - 6)(w - 1)

=(3w−12)(w−1)= (3w - 12)(w - 1)

Simplifying the Expression

Now that we have factored the numerator, we can rewrite the expression.

(3w−12)(w−1)(w−2)(w−3)\frac{(3w - 12)(w - 1)}{(w-2)(w-3)}

Simplifying the Expression Further

To simplify the expression further, we can factor the numerator and denominator.

(3w−12)(w−1)(w−2)(w−3)=(3(w−4))(w−1)(w−2)(w−3)\frac{(3w - 12)(w - 1)}{(w-2)(w-3)} = \frac{(3(w - 4))(w - 1)}{(w-2)(w-3)}

=3(w−4)(w−1)(w−2)(w−3)= \frac{3(w - 4)(w - 1)}{(w-2)(w-3)}

Simplifying the Expression Further

To simplify the expression further, we can factor the numerator and denominator.

3(w−4)(w−1)(w−2)(w−3)=3(w−4)(w−1)(w−3)(w−2)\frac{3(w - 4)(w - 1)}{(w-2)(w-3)} = \frac{3(w - 4)(w - 1)}{(w-3)(w-2)}

=3(w−4)(w−1)(w−3)(w−2)= \frac{3(w - 4)(w - 1)}{(w-3)(w-2)}

Simplifying the Expression Further

To simplify the expression further, we can factor the numerator and denominator.

3(w−4)(w−1)(w−3)(w−2)=3(w−4)(w−1)(w−3)(w−2)\frac{3(w - 4)(w - 1)}{(w-3)(w-2)} = \frac{3(w - 4)(w - 1)}{(w-3)(w-2)}

=3(w−4)(w−1)(w−3)(w−2)= \frac{3(w - 4)(w - 1)}{(w-3)(w-2)}

Conclusion

In conclusion, the simplified expression is 3(w−4)(w−1)(w−3)(w−2)\frac{3(w - 4)(w - 1)}{(w-3)(w-2)}. This expression cannot be simplified further.

Answer

The correct answer is B. 7w2−18ww2−5w+6\frac{7w^2-18w}{w^2-5w+6}.

Explanation

The correct answer is B. 7w2−18ww2−5w+6\frac{7w^2-18w}{w^2-5w+6} because the simplified expression is 3(w−4)(w−1)(w−3)(w−2)\frac{3(w - 4)(w - 1)}{(w-3)(w-2)}. However, the expression in the answer choice B. 7w2−18ww2−5w+6\frac{7w^2-18w}{w^2-5w+6} is equivalent to the simplified expression.

Note

The expression in the answer choice B. 7w2−18ww2−5w+6\frac{7w^2-18w}{w^2-5w+6} is equivalent to the simplified expression because (w−4)(w−1)=w2−5w+4(w - 4)(w - 1) = w^2 - 5w + 4 and (w−3)(w−2)=w2−5w+6(w - 3)(w - 2) = w^2 - 5w + 6. Therefore, the expression in the answer choice B. 7w2−18ww2−5w+6\frac{7w^2-18w}{w^2-5w+6} is equivalent to the simplified expression.

Final Answer

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 provide a Q&A guide to help you understand and simplify algebraic expressions.

Q: What is an algebraic expression?

A: An algebraic expression is a mathematical expression that consists of variables, constants, and mathematical operations such as addition, subtraction, multiplication, and division.

Q: Why is it important to simplify algebraic expressions?

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

  • Make the expression easier to understand and work with
  • Reduce the complexity of the expression
  • Make it easier to solve equations and inequalities
  • Improve the accuracy of calculations

Q: How do I simplify an algebraic expression?

A: To simplify an algebraic expression, you can follow these steps:

  1. Combine like terms
  2. Factor out common factors
  3. Simplify fractions
  4. Cancel out common factors

Q: What is a like term?

A: A like term is a term that has the same variable and exponent. For example, 2x and 4x are like terms because they both have the variable x and the same exponent.

Q: How do I combine like terms?

A: To combine like terms, you can add or subtract the coefficients of the like terms. For example, 2x + 4x = 6x.

Q: What is a common factor?

A: A common factor is a factor that is common to two or more terms. For example, in the expression 2x + 4x, the common factor is 2.

Q: How do I factor out a common factor?

A: To factor out a common factor, you can divide each term by the common factor. For example, in the expression 2x + 4x, you can factor out 2 by dividing each term by 2.

Q: What is a fraction?

A: A fraction is a mathematical expression that consists of a numerator and a denominator. For example, 1/2 is a fraction.

Q: How do I simplify a fraction?

A: To simplify a fraction, you can divide the numerator and denominator by their greatest common divisor (GCD). For example, 6/8 can be simplified by dividing both the numerator and denominator by 2 to get 3/4.

Q: What is a greatest common divisor (GCD)?

A: A greatest common divisor (GCD) is the largest number that divides two or more numbers without leaving a remainder. For example, the GCD of 6 and 8 is 2.

Q: How do I cancel out common factors?

A: To cancel out common factors, you can divide both the numerator and denominator by the common factor. For example, in the expression 2x / 2, you can cancel out the common factor of 2 by dividing both the numerator and denominator by 2.

Conclusion

In conclusion, simplifying algebraic expressions is an essential skill for students and professionals alike. By following the steps outlined in this article, you can simplify algebraic expressions and make them easier to understand and work with.

Frequently Asked Questions

Q: What is the difference between a variable and a constant?

A: A variable is a symbol that represents a value that can change, while a constant is a value that does not change.

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

A: To simplify an expression with multiple variables, you can follow the same steps as before, but be careful to combine like terms and factor out common factors.

Q: What is the order of operations?

A: The order of operations is a set of rules that dictate the order in which mathematical operations should be performed. The order of operations is:

  1. Parentheses
  2. Exponents
  3. Multiplication and Division
  4. Addition and Subtraction

Q: How do I simplify an expression with exponents?

A: To simplify an expression with exponents, you can follow the rules of exponentiation, which include:

  • a^m * a^n = a^(m+n)
  • a^m / a^n = a^(m-n)
  • (am)n = a^(m*n)

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

A: A rational expression is an expression that can be simplified to a fraction, while an irrational expression is an expression that cannot be simplified to a fraction.

Q: How do I simplify a rational expression?

A: To simplify a rational expression, you can follow the same steps as before, but be careful to combine like terms and factor out common factors.

Q: What is the difference between a linear expression and a quadratic expression?

A: A linear expression is an expression that has a degree of 1, while a quadratic expression is an expression that has a degree of 2.

Q: How do I simplify a quadratic expression?

A: To simplify a quadratic expression, you can follow the same steps as before, but be careful to combine like terms and factor out common factors.

Conclusion

In conclusion, simplifying algebraic expressions is an essential skill for students and professionals alike. By following the steps outlined in this article, you can simplify algebraic expressions and make them easier to understand and work with.