Subtract: \left(y^3 + 2v\right) - \left(-6y^3\right ]Your Answer Should Be In Simplest Terms. Enter The Correct Answer.

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Understanding the Problem

When dealing with algebraic expressions, it's essential to simplify them to make calculations easier and more manageable. In this article, we'll focus on simplifying a specific expression by subtracting one term from another. Our goal is to arrive at the simplest form of the given expression.

The Expression to Simplify

The expression we need to simplify is:

(y3+2v)−(−6y3)\left(y^3 + 2v\right) - \left(-6y^3\right)

Step 1: Distribute the Negative Sign

To simplify the expression, we'll start by distributing the negative sign to the terms inside the second set of parentheses. This will change the signs of all the terms inside the parentheses.

(y3+2v)−(−6y3)=(y3+2v)+6y3\left(y^3 + 2v\right) - \left(-6y^3\right) = \left(y^3 + 2v\right) + 6y^3

Step 2: Combine Like Terms

Now that we have distributed the negative sign, we can combine like terms. In this case, we have two terms with the same variable, y3y^3. We can add these terms together.

(y3+2v)+6y3=y3+6y3+2v\left(y^3 + 2v\right) + 6y^3 = y^3 + 6y^3 + 2v

Step 3: Simplify the Expression

We can simplify the expression further by combining the like terms. In this case, we have two terms with the same variable, y3y^3. We can add these terms together.

y3+6y3+2v=7y3+2vy^3 + 6y^3 + 2v = 7y^3 + 2v

The Final Answer

The simplified expression is:

7y3+2v7y^3 + 2v

Conclusion

Simplifying algebraic expressions is an essential skill in mathematics. By following the steps outlined in this article, we can simplify complex expressions and arrive at their simplest form. Remember to distribute the negative sign and combine like terms to simplify expressions.

Tips and Tricks

  • When simplifying expressions, always look for like terms and combine them.
  • Distribute the negative sign to the terms inside the second set of parentheses.
  • Use the order of operations (PEMDAS) to simplify expressions.

Common Mistakes

  • Failing to distribute the negative sign to the terms inside the second set of parentheses.
  • Not combining like terms.
  • Not following the order of operations (PEMDAS).

Real-World Applications

Simplifying algebraic expressions has numerous real-world applications. For example, in physics, we use algebraic expressions to describe the motion of objects. In engineering, we use algebraic expressions to design and optimize systems. In economics, we use algebraic expressions to model and analyze economic systems.

Final Thoughts

Simplifying algebraic expressions is a crucial skill in mathematics. By following the steps outlined in this article, we can simplify complex expressions and arrive at their simplest form. Remember to distribute the negative sign and combine like terms to simplify expressions. With practice and patience, you'll become proficient in simplifying algebraic expressions and tackle complex problems with confidence.

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Frequently Asked Questions

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

A: The first step in simplifying an algebraic expression is to distribute the negative sign to the terms inside the second set of parentheses.

Q: How do I combine like terms in an algebraic expression?

A: To combine like terms, look for terms with the same variable and coefficient. Add or subtract the coefficients of these terms to simplify the expression.

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

A: The order of operations (PEMDAS) is a set of rules that tells us which operations to perform first when simplifying 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 an expression with multiple variables?

A: To simplify an expression with multiple variables, follow the same steps as before: distribute the negative sign, combine like terms, and apply the order of operations (PEMDAS).

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

A: A variable is a letter or symbol that represents a value that can change. A constant is a value that does not change.

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

A: To simplify an expression with a negative coefficient, distribute the negative sign to the terms inside the second set of parentheses and then combine like terms.

Q: What is the final step in simplifying an algebraic expression?

A: The final step in simplifying an algebraic expression is to check your work and make sure that the expression is in its simplest form.

Common Misconceptions

Q: Do I need to simplify expressions with variables?

A: Yes, you should always simplify expressions with variables. Simplifying expressions helps to make calculations easier and more manageable.

Q: Can I simplify expressions with fractions?

A: Yes, you can simplify expressions with fractions by following the same steps as before: distribute the negative sign, combine like terms, and apply the order of operations (PEMDAS).

Q: Do I need to simplify expressions with exponents?

A: Yes, you should always simplify expressions with exponents. Simplifying expressions with exponents helps to make calculations easier and more manageable.

Real-World Applications

Q: How do algebraic expressions apply to real-world problems?

A: Algebraic expressions are used to model and analyze real-world problems in fields such as physics, engineering, and economics.

Q: Can I use algebraic expressions to solve problems in my daily life?

A: Yes, you can use algebraic expressions to solve problems in your daily life. For example, you can use algebraic expressions to calculate the cost of a product, the time it takes to complete a task, or the distance between two points.

Final Thoughts

Simplifying algebraic expressions is a crucial skill in mathematics. By following the steps outlined in this article, you can simplify complex expressions and arrive at their simplest form. Remember to distribute the negative sign, combine like terms, and apply the order of operations (PEMDAS) to simplify expressions. With practice and patience, you'll become proficient in simplifying algebraic expressions and tackle complex problems with confidence.