Simplify The Following Expression: X Y − 7 Xy^{-7} X Y − 7

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 expression $xy^{-7}$, which involves understanding the properties of exponents and variables. By the end of this discussion, you will have a clear understanding of how to simplify complex algebraic expressions.

Understanding Exponents

Exponents are a shorthand way of representing repeated multiplication. For example, $x^3$ means $x \times x \times x$. When dealing with negative exponents, we need to understand that they represent the reciprocal of the base raised to the positive exponent. In other words, $x^{-n}$ means $\frac{1}{x^n}$.

Simplifying the Expression $xy^{-7}$

To simplify the expression $xy^{-7}$, we need to apply the rules of exponents. Since the exponent is negative, we can rewrite it as a fraction:

$$xy^{-7} = x \times \frac{1}{y^7}$$

Now, we can simplify the expression by combining the variables:

$$x \times \frac{1}{y^7} = \frac{x}{y^7}$$

Properties of Exponents

When dealing with exponents, it's essential to understand the properties of multiplication and division. Here are some key properties to keep in mind:

• Product of Powers: When multiplying two powers with the same base, we add the exponents. For example, $x^m \times x^n = x^{m+n}$.
• Power of a Power: When raising a power to another power, we multiply the exponents. For example, $(x^m)^n = x^{mn}$.
• Quotient of Powers: When dividing two powers with the same base, we subtract the exponents. For example, $\frac{x^m}{x^n} = x^{m-n}$.

Applying the Properties of Exponents

Now that we have a simplified expression, $\frac{x}{y^7}$, we can apply the properties of exponents to further simplify it. Since the exponent is negative, we can rewrite it as a fraction:

$$\frac{x}{y^7} = x \times \frac{1}{y^7}$$

Using the property of the product of powers, we can rewrite the expression as:

$$x \times \frac{1}{y^7} = \frac{x}{y^7}$$

However, we can simplify the expression further by applying the property of the power of a power:

$$(y^{-7})^1 = y^{-7}$$

Using this property, we can rewrite the expression as:

$$\frac{x}{y^7} = \frac{x}{(y^7)^1}$$

Now, we can simplify the expression by applying the property of the power of a power:

$$\frac{x}{(y^7)^1} = \frac{x}{y^{7 \times 1}}$$

Using this property, we can rewrite the expression as:

$$\frac{x}{y^{7 \times 1}} = \frac{x}{y^7}$$

Conclusion

Simplifying algebraic expressions is an essential skill for students and professionals alike. By understanding the properties of exponents and variables, we can simplify complex expressions like $xy^{-7}$. In this article, we have applied the properties of exponents to simplify the expression, resulting in the final simplified form: $\frac{x}{y^7}$. By following these steps, you can simplify any algebraic expression and gain a deeper understanding of the underlying mathematics.

Common Mistakes to Avoid

When simplifying algebraic expressions, it's essential to avoid common mistakes. Here are some common pitfalls to watch out for:

• Incorrect application of exponent rules: Make sure to apply the correct exponent rules when simplifying expressions.
• Failure to simplify fractions: Don't forget to simplify fractions when possible.
• Incorrect order of operations: Make sure to follow the order of operations (PEMDAS) when simplifying expressions.

Practice Problems

To reinforce your understanding of simplifying algebraic expressions, try the following practice problems:

1. Simplify the expression $x^3y^{-2}$.
2. Simplify the expression $\frac{x^2}{y^4}$.
3. Simplify the expression $(x^3y^2)^{-1}$.

Additional Resources

For further practice and review, check out the following resources:

• Khan Academy: Algebra
• Mathway: Algebra
• Wolfram Alpha: Algebra

Introduction

In our previous article, we explored the concept of simplifying algebraic expressions, focusing on the expression $xy^{-7}$. We applied the properties of exponents and variables to simplify the expression, resulting in the final simplified form: $\frac{x}{y^7}$. In this article, we will address some common questions and concerns related to simplifying algebraic expressions.

Q&A

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

A: A variable is a symbol that represents a value that can change, such as $x$ or $y$. A constant is a value that does not change, such as $2$ or $5$.

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

A: To simplify an expression with multiple variables, you need to apply the properties of exponents and variables. For example, if you have the expression $x^2y^3$, you can simplify it by combining the variables: $x^2y^3 = (xy)^3$.

Q: What is the rule for multiplying powers with the same base?

A: When multiplying powers with the same base, you add the exponents. For example, $x^m \times x^n = x^{m+n}$.

Q: How do I simplify an expression with a negative exponent?

A: To simplify an expression with a negative exponent, you can rewrite it as a fraction. For example, $x^{-n} = \frac{1}{x^n}$.

Q: What is the rule for dividing powers with the same base?

A: When dividing powers with the same base, you subtract the exponents. For example, $\frac{x^m}{x^n} = x^{m-n}$.

Q: How do I simplify an expression with a fraction?

A: To simplify an expression with a fraction, you need to apply the properties of exponents and variables. For example, if you have the expression $\frac{x^2}{y^3}$, you can simplify it by combining the variables: $\frac{x^2}{y^3} = \frac{x^2}{(y^3)^1}$.

Q: What is the order of operations for simplifying expressions?

A: The order of operations for simplifying expressions is:

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

Common Mistakes to Avoid

When simplifying algebraic expressions, it's essential to avoid common mistakes. Here are some common pitfalls to watch out for:

• Incorrect application of exponent rules: Make sure to apply the correct exponent rules when simplifying expressions.
• Failure to simplify fractions: Don't forget to simplify fractions when possible.
• Incorrect order of operations: Make sure to follow the order of operations (PEMDAS) when simplifying expressions.

Practice Problems

To reinforce your understanding of simplifying algebraic expressions, try the following practice problems:

1. Simplify the expression $x^3y^{-2}$.
2. Simplify the expression $\frac{x^2}{y^4}$.
3. Simplify the expression $(x^3y^2)^{-1}$.

Additional Resources

For further practice and review, check out the following resources:

• Khan Academy: Algebra
• Mathway: Algebra
• Wolfram Alpha: Algebra

By following these steps and practicing with the provided resources, you'll become proficient in simplifying algebraic expressions and gain a deeper understanding of the underlying mathematics.