Simplifying Algebraic Expressions: A Detailed Guide

Hey guys! Ever get tangled up in simplifying algebraic expressions? It's like trying to untangle a ball of yarn, right? But don't worry, we're going to walk through it together. Let's break down this expression: $2 m \sqrt{3}-5 m \sqrt{2}+5$.

Understanding the Expression

Before we dive into simplifying, let's understand what we're looking at. This expression has three terms: $2m\sqrt{3}$, $-5m\sqrt{2}$, and $+5$. Each term is separated by addition or subtraction. The variables and constants are all mixed up, but we're here to bring some order to the chaos.

Breaking Down the Terms

1. $2m\sqrt{3}$: This term includes a coefficient (2), a variable (m), and a square root ($\sqrt{3}$). It means '2 times m times the square root of 3'.
2. $-5m\sqrt{2}$: Similar to the first term, this one has a coefficient (-5), a variable (m), and a square root ($\sqrt{2}$). It represents '-5 times m times the square root of 2'.
3. $+5$: This is a constant term. It's just the number 5.

Now that we know what each part means, we can start thinking about how to simplify the whole expression. The key thing to remember is that we can only combine like terms. Like terms have the same variable raised to the same power. In this case, the terms with 'm' also have different square roots, so they are not like terms.

Identifying Like Terms

So, what are 'like terms' anyway? Think of it like sorting socks. You can only pair up socks that are the same color and type. In algebra, like terms are terms that have the same variable raised to the same power. For example, $3x$ and $5x$ are like terms because they both have the variable 'x' raised to the power of 1. Similarly, $2x^2$ and $-7x^2$ are like terms because they both have the variable 'x' raised to the power of 2.

Why Like Terms Matter

Like terms matter because they are the only terms you can combine through addition or subtraction. It's like saying you can add apples to apples, but you can't directly add apples to oranges. So, $3x + 5x$ becomes $8x$, but $3x + 5y$ stays as $3x + 5y$ because 'x' and 'y' are different variables.

Looking at Our Expression

In our expression, $2 m \sqrt{3}-5 m \sqrt{2}+5$, let's identify the terms: $2m\sqrt{3}$, $-5m\sqrt{2}$, and $5$. Do we have any like terms here? Well:

• $2m\sqrt{3}$ and $-5m\sqrt{2}$ both have the variable 'm', but they have different square roots ($\sqrt{3}$ and $\sqrt{2}$). So, they are not like terms.
• The constant term $5$ doesn't have any 'm' in it, so it can't be combined with the other terms.

Since there are no like terms, we can't simplify the expression any further.

Simplifying the Expression

When we talk about simplifying algebraic expressions, we're essentially trying to make them as neat and compact as possible. This usually involves combining like terms, factoring, or using the order of operations (PEMDAS/BODMAS).

Combining Like Terms

As we discussed, combining like terms is a fundamental part of simplifying. Let's say we have the expression $3x + 2y - x + 4y$. To simplify this, we combine the 'x' terms and the 'y' terms:

• $3x - x = 2x$
• $2y + 4y = 6y$

So, the simplified expression is $2x + 6y$.

Factoring

Factoring is another powerful technique. It involves breaking down an expression into its factors. For example, if we have $6x + 9y$, we can factor out a common factor of 3:

• $6x + 9y = 3(2x + 3y)$

Order of Operations

Remember PEMDAS/BODMAS (Parentheses/Brackets, Exponents/Orders, Multiplication and Division, Addition and Subtraction)? Always follow this order when simplifying expressions. For instance, in the expression $2(3 + 4)^2 - 5$, we would first solve the parentheses, then the exponent, then multiplication, and finally subtraction:

• $2(3 + 4)^2 - 5 = 2(7)^2 - 5$
• $= 2(49) - 5$
• = $98 - 5$
• = $93$

Why Can't We Simplify Further?

In our original expression, $2 m \sqrt{3}-5 m \sqrt{2}+5$, we've already established that there are no like terms. This means we can't combine any of the terms through addition or subtraction. There's also no common factor we can factor out.

• The term $2m\sqrt{3}$ has variables $m$ and $\sqrt{3}$.
• The term $-5m\sqrt{2}$ has variables $m$ and $\sqrt{2}$.
• The term $5$ is a constant.

Since the square roots are different ($\sqrt{3}$ and $\sqrt{2}$), the terms $2m\sqrt{3}$ and $-5m\sqrt{2}$ are not like terms. The constant term $5$ is also not a like term with the other two. Therefore, we cannot simplify the expression any further.

Final Answer

Given the expression $2 m \sqrt{3}-5 m \sqrt{2}+5$, and after careful examination, we find that there are no like terms to combine and no common factors to extract. Thus, the expression is already in its simplest form. Therefore, the simplified expression remains: $2 m \sqrt{3}-5 m \sqrt{2}+5$.

So, the correct answer is:

D. $2 m \sqrt{3}-5 m \sqrt{2}+5$

Keep practicing, and you'll become a pro at simplifying expressions in no time! You got this!