Simplify The Expression:${ 10m + 12\left(\frac{2}{3}a - \frac{5}{6}m\right) }$

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 the given expression: $10m + 12\left(\frac{2}{3}a - \frac{5}{6}m\right)$. We will break down the expression into manageable parts, apply the rules of algebra, and arrive at a simplified form.

Understanding the Expression

The given expression is a combination of two terms: $10m$ and $12\left(\frac{2}{3}a - \frac{5}{6}m\right)$. The first term is a simple variable term, while the second term is a product of a constant and a difference of two terms. To simplify the expression, we need to apply the distributive property and combine like terms.

Applying the Distributive Property

The distributive property states that for any real numbers $a$, $b$, and $c$, $a(b + c) = ab + ac$. We can apply this property to the second term in the given expression: $12\left(\frac{2}{3}a - \frac{5}{6}m\right)$. By distributing the constant $12$ to each term inside the parentheses, we get:

$$12\left(\frac{2}{3}a - \frac{5}{6}m\right) = 12\left(\frac{2}{3}a\right) - 12\left(\frac{5}{6}m\right)$$

Simplifying the Expression

Now that we have applied the distributive property, we can simplify the expression further. We can start by simplifying each term separately:

$$12\left(\frac{2}{3}a\right) = 8a$$

$$12\left(\frac{5}{6}m\right) = 10m$$

Combining Like Terms

Now that we have simplified each term, we can combine like terms to arrive at the final simplified expression. We can combine the two terms $8a$ and $10m$ to get:

$$8a + 10m$$

Conclusion

In this article, we have simplified the given expression $10m + 12\left(\frac{2}{3}a - \frac{5}{6}m\right)$ by applying the distributive property and combining like terms. We have arrived at the final simplified expression: $8a + 10m$. This expression is a fundamental concept in algebra, and understanding how to simplify it is essential for solving complex mathematical problems.

Tips and Tricks

• When simplifying expressions, always start by applying the distributive property to any terms that are multiplied by a constant.
• Combine like terms to simplify the expression further.
• Use the rules of algebra to simplify the expression, such as combining fractions and simplifying exponents.

Real-World Applications

Simplifying expressions is a fundamental skill that has numerous real-world applications. In mathematics, simplifying expressions is essential for solving complex problems in algebra, geometry, and calculus. In science and engineering, simplifying expressions is crucial for modeling real-world phenomena and making predictions.

Common Mistakes to Avoid

When simplifying expressions, there are several common mistakes to avoid:

• Failing to apply the distributive property to terms that are multiplied by a constant.
• Not combining like terms to simplify the expression further.
• Making errors when simplifying fractions or exponents.

Final Thoughts

Simplifying expressions is a fundamental concept in mathematics that has numerous real-world applications. By understanding how to simplify expressions, we can solve complex mathematical problems and make predictions in science and engineering. In this article, we have simplified the given expression $10m + 12\left(\frac{2}{3}a - \frac{5}{6}m\right)$ by applying the distributive property and combining like terms. We have arrived at the final simplified expression: $8a + 10m$.

Introduction

In our previous article, we simplified the expression $10m + 12\left(\frac{2}{3}a - \frac{5}{6}m\right)$ by applying the distributive property and combining like terms. We arrived at the final simplified expression: $8a + 10m$. In this article, we will answer some frequently asked questions related to simplifying expressions.

Q&A

Q: What is the distributive property, and how is it used in simplifying expressions?

A: The distributive property is a fundamental concept in algebra that states that for any real numbers $a$, $b$, and $c$, $a(b + c) = ab + ac$. It is used to simplify expressions by distributing a constant to each term inside the parentheses.

Q: How do I apply the distributive property to simplify an expression?

A: To apply the distributive property, you need to multiply the constant outside the parentheses to each term inside the parentheses. For example, if you have the expression $12\left(\frac{2}{3}a - \frac{5}{6}m\right)$, you would multiply $12$ to each term inside the parentheses to get $8a - 10m$.

Q: What are like terms, and how do I combine them to simplify an expression?

A: Like terms are terms that have the same variable and exponent. To combine like terms, you need to add or subtract the coefficients of the like terms. For example, if you have the expression $3x + 2x$, you would combine the like terms to get $5x$.

Q: How do I simplify fractions in an expression?

A: To simplify fractions, you need to find the greatest common divisor (GCD) of the numerator and denominator and divide both by the GCD. For example, if you have the expression $\frac{6}{8}$, you would simplify it to $\frac{3}{4}$ by dividing both the numerator and denominator by $2$.

Q: What are some common mistakes to avoid when simplifying expressions?

A: Some common mistakes to avoid when simplifying expressions include:

• Failing to apply the distributive property to terms that are multiplied by a constant.
• Not combining like terms to simplify the expression further.
• Making errors when simplifying fractions or exponents.

Q: How do I check if an expression is simplified?

A: To check if an expression is simplified, you need to make sure that:

• All like terms have been combined.
• All fractions have been simplified.
• All exponents have been simplified.

Tips and Tricks

• Always start by applying the distributive property to any terms that are multiplied by a constant.
• Combine like terms to simplify the expression further.
• Use the rules of algebra to simplify the expression, such as combining fractions and simplifying exponents.

Real-World Applications

Simplifying expressions is a fundamental skill that has numerous real-world applications. In mathematics, simplifying expressions is essential for solving complex problems in algebra, geometry, and calculus. In science and engineering, simplifying expressions is crucial for modeling real-world phenomena and making predictions.

Conclusion

In this article, we have answered some frequently asked questions related to simplifying expressions. We have covered topics such as the distributive property, like terms, and simplifying fractions. We have also provided some tips and tricks for simplifying expressions and highlighted the importance of this skill in real-world applications.

Final Thoughts

Simplifying expressions is a fundamental concept in mathematics that has numerous real-world applications. By understanding how to simplify expressions, we can solve complex mathematical problems and make predictions in science and engineering. In this article, we have provided a comprehensive guide to simplifying expressions, including tips and tricks and real-world applications.