Determine Which Expression(s) Is In Its Simplified Form. Select All Situations That Apply.A. { \frac{x^2}{xy}$}$B. { \frac{4x}{32z}$}$C. { \frac{12x}{y}$}$D. { \frac{z^4}{xyz}$}$E. { \frac{5z}{xy}$}$

Introduction

Simplifying algebraic expressions is a crucial skill in mathematics, as it allows us to rewrite complex expressions in a more manageable form. This, in turn, enables us to perform calculations and solve equations more efficiently. In this article, we will explore the concept of simplifying algebraic expressions and determine which of the given expressions are in their simplified form.

What is a Simplified Algebraic Expression?

A simplified algebraic expression is one that has been rewritten in a way that eliminates any unnecessary complexity. This is typically achieved by combining like terms, canceling out common factors, and rearranging the expression to make it easier to work with.

Key Concepts

Before we dive into the examples, let's review some key concepts that are essential for simplifying algebraic expressions:

• Like terms: These are terms that have the same variable(s) raised to the same power. For example, 2x and 3x are like terms.
• Common factors: These are factors that appear in both the numerator and denominator of an expression. For example, in the expression $\frac{6x}{2x}$, the common factor is 2x.
• Canceling out common factors: This involves dividing both the numerator and denominator by a common factor to simplify the expression.

Example A: $\frac{x^2}{xy}$

Let's start by analyzing the first expression: $\frac{x^2}{xy}$. To simplify this expression, we can cancel out the common factor of x from the numerator and denominator:

$$\frac{x^2}{xy} = \frac{x \cdot x}{x \cdot y} = \frac{x}{y}$$

Therefore, the simplified form of expression A is $\frac{x}{y}$.

Example B: $\frac{4x}{32z}$

Next, let's consider the second expression: $\frac{4x}{32z}$. To simplify this expression, we can cancel out the common factor of 4 from the numerator and denominator:

$$\frac{4x}{32z} = \frac{1 \cdot x}{8 \cdot z} = \frac{x}{8z}$$

However, we can further simplify this expression by canceling out the common factor of 8 from the numerator and denominator:

$$\frac{x}{8z} = \frac{x}{8 \cdot z} = \frac{x}{8z}$$

Therefore, the simplified form of expression B is $\frac{x}{8z}$.

Example C: $\frac{12x}{y}$

Now, let's analyze the third expression: $\frac{12x}{y}$. To simplify this expression, we can cancel out the common factor of 12 from the numerator and denominator:

$$\frac{12x}{y} = \frac{3 \cdot 4 \cdot x}{y} = \frac{3 \cdot 4x}{y}$$

However, we can further simplify this expression by canceling out the common factor of 3 from the numerator and denominator:

$$\frac{3 \cdot 4x}{y} = \frac{3 \cdot 4 \cdot x}{y} = \frac{3 \cdot 4x}{y}$$

Therefore, the simplified form of expression C is $\frac{3 \cdot 4x}{y}$.

Example D: $\frac{z^4}{xyz}$

Next, let's consider the fourth expression: $\frac{z^4}{xyz}$. To simplify this expression, we can cancel out the common factor of z from the numerator and denominator:

$$\frac{z^4}{xyz} = \frac{z \cdot z \cdot z \cdot z}{x \cdot y \cdot z} = \frac{z^3}{xy}$$

Therefore, the simplified form of expression D is $\frac{z^3}{xy}$.

Example E: $\frac{5z}{xy}$

Finally, let's analyze the fifth expression: $\frac{5z}{xy}$. To simplify this expression, we can cancel out the common factor of 5 from the numerator and denominator:

$$\frac{5z}{xy} = \frac{5 \cdot z}{x \cdot y} = \frac{5z}{xy}$$

However, we can further simplify this expression by canceling out the common factor of 5 from the numerator and denominator:

$$\frac{5z}{xy} = \frac{5 \cdot z}{x \cdot y} = \frac{5z}{xy}$$

Therefore, the simplified form of expression E is $\frac{5z}{xy}$.

Conclusion

In conclusion, the simplified forms of the given expressions are:

• Expression A: $\frac{x}{y}$
• Expression B: $\frac{x}{8z}$
• Expression C: $\frac{3 \cdot 4x}{y}$
• Expression D: $\frac{z^3}{xy}$
• Expression E: $\frac{5z}{xy}$

Introduction

In our previous article, we explored the concept of simplifying algebraic expressions and determined which of the given expressions are in their simplified form. In this article, we will answer some frequently asked questions about simplifying algebraic expressions.

Q: What is the first step in simplifying an algebraic expression?

A: The first step in simplifying an algebraic expression is to look for common factors in the numerator and denominator. Common factors are factors that appear in both the numerator and denominator of an expression.

Q: How do I identify like terms in an algebraic expression?

A: Like terms are terms that have the same variable(s) raised to the same power. For example, 2x and 3x are like terms because they both have the variable x raised to the power of 1.

Q: Can I simplify an algebraic expression by canceling out a common factor that appears only in the numerator or denominator?

A: No, you cannot simplify an algebraic expression by canceling out a common factor that appears only in the numerator or denominator. To simplify an expression, you must cancel out a common factor that appears in both the numerator and denominator.

Q: How do I know if an algebraic expression is in its simplified form?

A: An algebraic expression is in its simplified form if it cannot be simplified further by canceling out common factors or combining like terms.

Q: Can I simplify an algebraic expression by rearranging the terms?

A: Yes, you can simplify an algebraic expression by rearranging the terms. However, this is not always necessary, and you should only rearrange the terms if it makes the expression easier to work with.

Q: What is the difference between simplifying an algebraic expression and factoring it?

A: Simplifying an algebraic expression involves canceling out common factors and combining like terms to make the expression easier to work with. Factoring an algebraic expression involves expressing it as a product of simpler expressions.

Q: Can I simplify an algebraic expression with variables in the denominator?

A: Yes, you can simplify an algebraic expression with variables in the denominator. However, you must be careful not to cancel out a variable that appears in the denominator.

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

A: To simplify an algebraic expression with multiple variables, you must look for common factors in the numerator and denominator and cancel them out. You must also combine like terms and rearrange the expression as necessary.

Q: Can I simplify an algebraic expression with fractions in the numerator and denominator?

A: Yes, you can simplify an algebraic expression with fractions in the numerator and denominator. However, you must be careful not to cancel out a fraction that appears in the denominator.

Conclusion

In conclusion, simplifying algebraic expressions is an essential skill in mathematics, and it requires a combination of techniques and strategies. By following the steps outlined in this article, you should be able to simplify algebraic expressions with ease. Remember to always look for common factors and like terms, and don't be afraid to cancel them out to simplify the expression.

Common Mistakes to Avoid

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

• Canceling out a common factor that appears only in the numerator or denominator
• Failing to combine like terms
• Rearranging the terms in a way that makes the expression more complicated
• Canceling out a variable that appears in the denominator
• Failing to simplify an expression with multiple variables

By avoiding these common mistakes, you can ensure that your algebraic expressions are simplified correctly and efficiently.

Practice Problems

To practice simplifying algebraic expressions, try the following problems:

1. Simplify the expression $\frac{2x^2}{4x}$.
2. Simplify the expression $\frac{3y^2}{6y}$.
3. Simplify the expression $\frac{4z^3}{8z}$.
4. Simplify the expression $\frac{5x^2}{10x}$.
5. Simplify the expression $\frac{6y^3}{12y}$.

By practicing these problems, you should be able to develop your skills in simplifying algebraic expressions and become more confident in your ability to work with these types of expressions.