{ \left(5x^3 - 3\right) - \left(-4x^3 + 8\right)$}$The Given Expression Is Equivalent To { Bx^3 - 11$}$, Where { B$}$ Is A Constant. What Is The Value Of { B$}$? { \square$}$

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Introduction

Algebraic expressions are a fundamental concept in mathematics, and simplifying them is an essential skill for any math enthusiast. In this article, we will explore the process of simplifying algebraic expressions, with a focus on the given expression {\left(5x^3 - 3\right) - \left(-4x^3 + 8\right)$}$. We will break down the expression into smaller parts, apply the rules of algebra, and finally arrive at the simplified form.

Understanding the Given Expression

The given expression is {\left(5x^3 - 3\right) - \left(-4x^3 + 8\right)$}$. This expression consists of two parts: ${5x^3 - 3\$} and {-4x^3 + 8$}$. The first part is a polynomial expression with a degree of 3, while the second part is also a polynomial expression with a degree of 3.

Applying the Rules of Algebra

To simplify the given expression, we need to apply the rules of algebra. The first rule is the distributive property, which states that for any real numbers {a$}$, {b$}$, and {c$}$, {a(b + c) = ab + ac$}$. The second rule is the commutative property, which states that for any real numbers {a$}$ and {b$}$, {a + b = b + a$}$.

Step 1: Distribute the Negative Sign

The first step is to distribute the negative sign to the terms inside the second set of parentheses. This will change the sign of each term inside the parentheses.

{\left(5x^3 - 3\right) - \left(-4x^3 + 8\right) = 5x^3 - 3 + 4x^3 - 8$}$

Step 2: Combine Like Terms

The next step is to combine like terms. Like terms are terms that have the same variable and exponent. In this case, we have two terms with the variable {x^3$}$: ${5x^3\$} and ${4x^3\$}. We can combine these terms by adding their coefficients.

${5x^3 + 4x^3 = 9x^3\$}

Now, we can rewrite the expression as:

${9x^3 - 3 - 8\$}

Step 3: Combine Constants

The final step is to combine the constants. In this case, we have two constants: {-3$}$ and {-8$}$. We can combine these constants by adding them.

{-3 + (-8) = -11$}$

Now, we can rewrite the expression as:

${9x^3 - 11\$}

Conclusion

In conclusion, the given expression {\left(5x^3 - 3\right) - \left(-4x^3 + 8\right)$}$ is equivalent to ${9x^3 - 11\$}. The value of {b$}$ is ${9\$}.

The Final Answer

The final answer is {\boxed{9}$}$.

Simplifying Algebraic Expressions: Tips and Tricks

Simplifying algebraic expressions can be a challenging task, but with practice and patience, you can master this skill. Here are some tips and tricks to help you simplify algebraic expressions:

  • Start by identifying like terms: Like terms are terms that have the same variable and exponent. Combining like terms is the key to simplifying algebraic expressions.
  • Use the distributive property: The distributive property states that for any real numbers {a$}$, {b$}$, and {c$}$, {a(b + c) = ab + ac$}$. This property can help you simplify expressions by distributing the terms inside the parentheses.
  • Combine constants: Constants are numbers that do not have a variable. Combining constants is an essential step in simplifying algebraic expressions.
  • Check your work: Simplifying algebraic expressions requires attention to detail. Make sure to check your work to ensure that you have not made any mistakes.

Introduction

Simplifying algebraic expressions is a fundamental concept in mathematics, and it can be a challenging task for many students. In our previous article, we explored the process of simplifying algebraic expressions, with a focus on the given expression {\left(5x^3 - 3\right) - \left(-4x^3 + 8\right)$}$. In this article, we will answer some frequently asked questions about simplifying algebraic expressions.

Q: What is the distributive property?

A: The distributive property is a fundamental concept in algebra that states that for any real numbers {a$}$, {b$}$, and {c$}$, {a(b + c) = ab + ac$}$. This property can help you simplify expressions by distributing the terms inside the parentheses.

Q: How do I identify like terms?

A: Like terms are terms that have the same variable and exponent. To identify like terms, look for terms that have the same variable and exponent. For example, ${2x^2\$} and ${3x^2\$} are like terms because they have the same variable and exponent.

Q: How do I combine like terms?

A: To combine like terms, add or subtract the coefficients of the like terms. For example, if you have $2x^2 + 3x^2\$}, you can combine the like terms by adding the coefficients ${$2 + 3 = 5$$. Therefore, ${2x^2 + 3x^2 = 5x^2\$}.

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

A: A constant is a number that does not have a variable. For example, ${5\$} is a constant because it does not have a variable. A variable, on the other hand, is a letter or symbol that represents a value that can change. For example, {x$}$ is a variable because it represents a value that can change.

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

A: To simplify an expression with multiple variables, follow the same steps as before: identify like terms, combine like terms, and simplify the expression. For example, if you have the expression $2x^2 + 3y^2 + 4x^2\$}, you can simplify it by combining like terms ${$2x^2 + 4x^2 = 6x^2$$.

Q: What is the order of operations?

A: The order of operations is a set of rules that tells you which operations to perform first when simplifying an expression. The order of operations is:

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

Q: How do I check my work?

A: To check your work, follow these steps:

  1. Simplify the expression using the order of operations.
  2. Check that you have combined like terms correctly.
  3. Check that you have simplified the expression correctly.
  4. Check that your final answer is correct.

By following these steps, you can ensure that your work is accurate and complete.

Conclusion

Simplifying algebraic expressions is a fundamental concept in mathematics, and it can be a challenging task for many students. By following the steps outlined in this article, you can simplify algebraic expressions with ease. Remember to practice regularly to develop your skills and become proficient in simplifying algebraic expressions.