Simplify The Expression:$\[ 7b^2 - 4b^2 - 6b^2 + 5b^2 \\]

by ADMIN 58 views

Introduction

Algebraic expressions are a fundamental concept in mathematics, and simplifying them is a crucial skill to master. In this article, we will focus on simplifying the given expression: 7b2−4b2−6b2+5b27b^2 - 4b^2 - 6b^2 + 5b^2. We will break down the process into manageable steps, making it easier to understand and apply.

Understanding the Expression

The given expression is a combination of four terms, each containing a variable bb raised to the power of 2. The expression can be written as:

7b2−4b2−6b2+5b27b^2 - 4b^2 - 6b^2 + 5b^2

Step 1: Combine Like Terms

The first step in simplifying the expression is to combine like terms. Like terms are terms that have the same variable raised to the same power. In this case, we have four terms with the variable bb raised to the power of 2.

To combine like terms, we need to add or subtract the coefficients of the terms. The coefficients are the numbers in front of the variables.

Let's combine the terms:

7b2−4b2−6b2+5b27b^2 - 4b^2 - 6b^2 + 5b^2

We can start by combining the first two terms:

7b2−4b2=3b27b^2 - 4b^2 = 3b^2

Now, we can combine the result with the third term:

3b2−6b2=−3b23b^2 - 6b^2 = -3b^2

Finally, we can combine the result with the fourth term:

−3b2+5b2=2b2-3b^2 + 5b^2 = 2b^2

Step 2: Simplify the Expression

Now that we have combined the like terms, we can simplify the expression by writing it in its simplest form.

The simplified expression is:

2b22b^2

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.

In this article, we simplified the expression 7b2−4b2−6b2+5b27b^2 - 4b^2 - 6b^2 + 5b^2 by combining like terms and writing it in its simplest form.

Tips and Tricks

Here are some tips and tricks to help you simplify algebraic expressions:

  • Combine like terms: Combine terms with the same variable raised to the same power.
  • Use the distributive property: Use the distributive property to expand expressions and simplify them.
  • Use the commutative property: Use the commutative property to rearrange terms and simplify expressions.
  • Use the associative property: Use the associative property to group terms and simplify expressions.

Real-World Applications

Simplifying algebraic expressions has many real-world applications. Here are a few examples:

  • Science: Simplifying algebraic expressions is essential in science, where complex equations need to be solved to understand natural phenomena.
  • Engineering: Simplifying algebraic expressions is crucial in engineering, where complex systems need to be analyzed and optimized.
  • Finance: Simplifying algebraic expressions is essential in finance, where complex financial models need to be analyzed and optimized.

Common Mistakes

Here are some common mistakes to avoid when simplifying algebraic expressions:

  • Not combining like terms: Failing to combine like terms can lead to incorrect simplifications.
  • Not using the distributive property: Failing to use the distributive property can lead to incorrect simplifications.
  • Not using the commutative property: Failing to use the commutative property can lead to incorrect simplifications.
  • Not using the associative property: Failing to use the associative property can lead to incorrect simplifications.

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.

In this article, we simplified the expression 7b2−4b2−6b2+5b27b^2 - 4b^2 - 6b^2 + 5b^2 by combining like terms and writing it in its simplest form.

We also discussed some tips and tricks to help you simplify algebraic expressions, as well as some common mistakes to avoid.

By mastering the art of simplifying algebraic expressions, you will be able to tackle complex mathematical problems with confidence and accuracy.

Final Answer

Introduction

In our previous article, we discussed the importance of simplifying algebraic expressions and provided a step-by-step guide on how to simplify the expression 7b2−4b2−6b2+5b27b^2 - 4b^2 - 6b^2 + 5b^2. In this article, we will answer some frequently asked questions (FAQs) related to simplifying algebraic expressions.

Q&A

Q: What is the difference between like terms and unlike terms?

A: Like terms are terms that have the same variable raised to the same power. Unlike terms are terms that have different variables or different powers of the same variable.

Q: How do I combine like terms?

A: To combine like terms, you need to add or subtract the coefficients of the terms. The coefficients are the numbers in front of the variables.

Q: What is the distributive property?

A: The distributive property is a mathematical property that allows you to expand expressions by multiplying each term inside the parentheses by the factor outside the parentheses.

Q: How do I use the distributive property to simplify expressions?

A: To use the distributive property, you need to multiply each term inside the parentheses by the factor outside the parentheses. For example, if you have the expression 2(x+3)2(x + 3), you can use the distributive property to expand it as 2x+62x + 6.

Q: What is the commutative property?

A: The commutative property is a mathematical property that allows you to rearrange terms in an expression without changing its value.

Q: How do I use the commutative property to simplify expressions?

A: To use the commutative property, you need to rearrange the terms in the expression to make it easier to simplify. For example, if you have the expression x+3yx + 3y, you can use the commutative property to rearrange it as 3y+x3y + x.

Q: What is the associative property?

A: The associative property is a mathematical property that allows you to group terms in an expression without changing its value.

Q: How do I use the associative property to simplify expressions?

A: To use the associative property, you need to group the terms in the expression to make it easier to simplify. For example, if you have the expression (x+2)+(3y+4)(x + 2) + (3y + 4), you can use the associative property to group it as (x+3y)+(2+4)(x + 3y) + (2 + 4).

Q: How do I simplify expressions with variables in the denominator?

A: To simplify expressions with variables in the denominator, you need to multiply both the numerator and the denominator by the conjugate of the denominator. The conjugate of a binomial expression a+ba + b is a−ba - b.

Q: What is the conjugate of a binomial expression?

A: The conjugate of a binomial expression a+ba + b is a−ba - b.

Q: How do I simplify expressions with fractions?

A: To simplify expressions with fractions, you need to find the least common multiple (LCM) of the denominators and multiply both the numerator and the denominator by the LCM.

Q: What is the least common multiple (LCM)?

A: The least common multiple (LCM) of two or more numbers is the smallest number that is a multiple of each of the numbers.

Q: How do I simplify expressions with exponents?

A: To simplify expressions with exponents, you need to apply the rules of exponents, such as the product rule and the power rule.

Q: What is the product rule for exponents?

A: The product rule for exponents states that when you multiply two numbers with the same base, you add their exponents.

Q: What is the power rule for exponents?

A: The power rule for exponents states that when you raise a number with an exponent to another power, you multiply the exponents.

Conclusion

Simplifying algebraic expressions is an essential skill in mathematics. By understanding the concepts and techniques discussed in this article, you will be able to simplify complex expressions and arrive at their simplest form.

In this article, we answered some frequently asked questions (FAQs) related to simplifying algebraic expressions. We discussed the difference between like terms and unlike terms, how to combine like terms, and how to use the distributive property, commutative property, and associative property to simplify expressions.

We also discussed how to simplify expressions with variables in the denominator, fractions, and exponents.

By mastering the art of simplifying algebraic expressions, you will be able to tackle complex mathematical problems with confidence and accuracy.

Final Answer

The final answer is: 2b2\boxed{2b^2}