Simplify The Expression:${ 6m - 3m^5 - M^2 + 12 }$

Introduction

In mathematics, simplifying expressions is a crucial skill that helps us to solve complex problems and understand the underlying concepts. In this article, we will focus on simplifying the given expression: $6m - 3m^5 - m^2 + 12$. We will break down the expression into smaller parts, identify the like terms, and combine them to simplify the expression.

Understanding the Expression

The given expression is a polynomial expression, which consists of several terms with different powers of the variable $m$. The expression is: $6m - 3m^5 - m^2 + 12$. To simplify this expression, we need to understand the properties of exponents and the rules of combining like terms.

Properties of Exponents

Exponents are a shorthand way of writing repeated multiplication. For example, $m^2$ means $m \times m$, and $m^3$ means $m \times m \times m$. When we multiply two or more terms with the same base, we add their exponents. For example, $m^2 \times m^3 = m^{2+3} = m^5$.

Like Terms

Like terms are terms that have the same variable and exponent. For example, $2m$ and $3m$ are like terms because they both have the variable $m$ and the exponent $1$. When we combine like terms, we add their coefficients. For example, $2m + 3m = 5m$.

Simplifying the Expression

Now that we have understood the properties of exponents and like terms, we can simplify the given expression. The expression is: $6m - 3m^5 - m^2 + 12$. We can start by identifying the like terms. The terms $6m$ and $-m^2$ are like terms because they both have the variable $m$ and the exponent $1$. The terms $-3m^5$ and $12$ are not like terms because they have different variables and exponents.

Step 1: Combine Like Terms

We can combine the like terms $6m$ and $-m^2$ by adding their coefficients. The coefficient of $6m$ is $6$, and the coefficient of $-m^2$ is $-1$. Therefore, the combined term is: $6m - m^2$.

Step 2: Rewrite the Expression

Now that we have combined the like terms, we can rewrite the expression. The expression is: $6m - m^2 - 3m^5 + 12$. We can rewrite this expression as: $-3m^5 - m^2 + 6m + 12$.

Step 3: Simplify the Expression

The expression $-3m^5 - m^2 + 6m + 12$ is already simplified. We have combined all the like terms, and there are no more terms to simplify.

Conclusion

In this article, we have simplified the expression: $6m - 3m^5 - m^2 + 12$. We have identified the like terms, combined them, and rewritten the expression. The final simplified expression is: $-3m^5 - m^2 + 6m + 12$. This expression is now in its simplest form, and we can use it to solve complex problems and understand the underlying concepts.

Final Answer

The final answer is: $-3m^5 - m^2 + 6m + 12$.

Additional Resources

For more information on simplifying expressions, please refer to the following resources:

• Khan Academy: Simplifying Expressions
• Mathway: Simplifying Expressions
• Wolfram Alpha: Simplifying Expressions

FAQs

Q: What is the difference between like terms and unlike terms? A: Like terms are terms that have the same variable and exponent, while unlike terms are terms that have different variables or exponents.

Q: How do I combine like terms? A: To combine like terms, you add their coefficients.

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Q&A: Simplifying Expressions

Q: What is the purpose of simplifying expressions? A: The purpose of simplifying expressions is to make them easier to work with and understand. Simplifying expressions helps us to identify patterns and relationships between variables, which is essential in solving complex problems.

Q: What are like terms? A: Like terms are terms that have the same variable and exponent. For example, $2m$ and $3m$ are like terms because they both have the variable $m$ and the exponent $1$.

Q: How do I identify like terms? A: To identify like terms, you need to look for terms that have the same variable and exponent. You can also use the distributive property to expand expressions and identify like terms.

Q: How do I combine like terms? A: To combine like terms, you add their coefficients. For example, $2m + 3m = 5m$.

Q: What is the difference between combining like terms and simplifying expressions? A: Combining like terms is a step in simplifying expressions. Simplifying expressions involves combining like terms, as well as removing any unnecessary parentheses or brackets.

Q: Can I simplify expressions with variables in the denominator? A: Yes, you can simplify expressions with variables in the denominator. However, you need to be careful when simplifying expressions with variables in the denominator, as you may need to use the rules of exponents and fractions.

Q: How do I simplify expressions with fractions? A: To simplify expressions with fractions, you need to find a common denominator and combine the fractions. You can also use the rules of exponents and fractions to simplify expressions with fractions.

Q: Can I simplify expressions with negative exponents? A: Yes, you can simplify expressions with negative exponents. However, you need to be careful when simplifying expressions with negative exponents, as you may need to use the rules of exponents and fractions.

Q: How do I simplify expressions with absolute values? A: To simplify expressions with absolute values, you need to remove the absolute value signs and simplify the expression inside the absolute value signs.

Q: Can I simplify expressions with radicals? A: Yes, you can simplify expressions with radicals. However, you need to be careful when simplifying expressions with radicals, as you may need to use the rules of radicals and exponents.

Q: How do I simplify expressions with complex numbers? A: To simplify expressions with complex numbers, you need to use the rules of complex numbers and exponents. You can also use the distributive property to expand expressions and simplify complex numbers.

Q: Can I simplify expressions with trigonometric functions? A: Yes, you can simplify expressions with trigonometric functions. However, you need to be careful when simplifying expressions with trigonometric functions, as you may need to use the rules of trigonometric functions and exponents.

Q: How do I simplify expressions with logarithmic functions? A: To simplify expressions with logarithmic functions, you need to use the rules of logarithmic functions and exponents. You can also use the distributive property to expand expressions and simplify logarithmic functions.

Q: Can I simplify expressions with exponential functions? A: Yes, you can simplify expressions with exponential functions. However, you need to be careful when simplifying expressions with exponential functions, as you may need to use the rules of exponential functions and exponents.

Q: How do I simplify expressions with polynomial functions? A: To simplify expressions with polynomial functions, you need to use the rules of polynomial functions and exponents. You can also use the distributive property to expand expressions and simplify polynomial functions.

Conclusion

Simplifying expressions is an essential skill in mathematics, and it requires a deep understanding of the rules of exponents, fractions, and other mathematical concepts. By following the steps outlined in this article, you can simplify expressions and solve complex problems with ease.

Additional Resources

For more information on simplifying expressions, please refer to the following resources:

• Khan Academy: Simplifying Expressions
• Mathway: Simplifying Expressions
• Wolfram Alpha: Simplifying Expressions

FAQs

Q: What is the difference between simplifying expressions and solving equations? A: Simplifying expressions involves combining like terms and removing unnecessary parentheses or brackets, while solving equations involves finding the value of a variable that makes the equation true.

Q: How do I know when to simplify an expression? A: You should simplify an expression when it is necessary to make the expression easier to work with or understand.

Q: Can I simplify expressions with variables in the numerator and denominator? A: Yes, you can simplify expressions with variables in the numerator and denominator. However, you need to be careful when simplifying expressions with variables in the numerator and denominator, as you may need to use the rules of exponents and fractions.

Q: How do I simplify expressions with multiple variables? A: To simplify expressions with multiple variables, you need to use the rules of exponents and fractions, as well as the distributive property to expand expressions.

Q: Can I simplify expressions with absolute values and radicals? A: Yes, you can simplify expressions with absolute values and radicals. However, you need to be careful when simplifying expressions with absolute values and radicals, as you may need to use the rules of absolute values and radicals.

Q: How do I simplify expressions with complex numbers and trigonometric functions? A: To simplify expressions with complex numbers and trigonometric functions, you need to use the rules of complex numbers and trigonometric functions, as well as the distributive property to expand expressions.

Q: Can I simplify expressions with logarithmic functions and exponential functions? A: Yes, you can simplify expressions with logarithmic functions and exponential functions. However, you need to be careful when simplifying expressions with logarithmic functions and exponential functions, as you may need to use the rules of logarithmic functions and exponential functions.