Evaluate: $ X Y - X \left( X - Y^0 \right) $

Mar 11, 2025 by ADMIN 45 views

$Evaluate: $ x y - x \left( x - y^0 \right) $$

Introduction

In mathematics, algebraic expressions are a fundamental concept that helps us solve various problems. Evaluating these expressions is a crucial step in solving equations and inequalities. In this article, we will evaluate the expression $ x y - x \left( x - y^0 \right) $ and provide a step-by-step solution.

Understanding the Expression

The given expression is $ x y - x \left( x - y^0 \right) $. To evaluate this expression, we need to follow the order of operations (PEMDAS):

Parentheses: Evaluate the expressions inside the parentheses.
Exponents: Evaluate any exponential expressions.
Multiplication and Division: Evaluate any multiplication and division operations from left to right.
Addition and Subtraction: Finally, evaluate any addition and subtraction operations from left to right.

Evaluating the Expression

Let's start by evaluating the expression inside the parentheses:

$ x - y^0 $

Since $ y^0 $ is equal to 1, we can simplify the expression as:

$ x - 1 $

Now, let's substitute this expression back into the original expression:

$ x y - x \left( x - 1 \right) $

Distributing the $ x $ Term

To simplify the expression, we need to distribute the $ x $ term:

$ x y - x x + x $

Combining Like Terms

Now, let's combine like terms:

$ x y - x^2 + x $

Final Evaluation

The final evaluation of the expression $ x y - x \left( x - y^0 \right) $ is:

$ x y - x^2 + x $

Conclusion

In this article, we evaluated the expression $ x y - x \left( x - y^0 \right) $ using the order of operations (PEMDAS). We simplified the expression by distributing the $ x $ term and combining like terms. The final evaluation of the expression is $ x y - x^2 + x $.

Step-by-Step Solution

Here is a step-by-step solution to evaluate the expression:

Evaluate the expression inside the parentheses: $ x - y^0 = x - 1 $
Substitute the expression back into the original expression: $ x y - x \left( x - 1 \right) $
Distribute the $ x $ term: $ x y - x x + x $
Combine like terms: $ x y - x^2 + x $

Frequently Asked Questions

Q: What is the order of operations (PEMDAS)?

A: The order of operations (PEMDAS) is a set of rules that helps us evaluate mathematical expressions. The acronym PEMDAS stands for:

Parentheses
Exponents
Multiplication and Division
Addition and Subtraction

Q: How do I evaluate an expression with parentheses?

A: To evaluate an expression with parentheses, we need to follow the order of operations (PEMDAS). First, evaluate the expressions inside the parentheses, then evaluate any exponential expressions, and finally evaluate any multiplication and division operations from left to right.

Q: What is the difference between $ y^0 $ and $ y^1 $?

A: The difference between $ y^0 $ and $ y^1 $ is that $ y^0 $ is equal to 1, while $ y^1 $ is equal to $ y $. For example, if $ y = 2 $, then $ y^0 = 1 $ and $ y^1 = 2 $.

Additional Resources

If you want to learn more about evaluating algebraic expressions, here are some additional resources:

Khan Academy: Algebraic Expressions
Mathway: Algebraic Expressions
Wolfram Alpha: Algebraic Expressions

Conclusion

Introduction

Evaluating algebraic expressions is a crucial step in solving equations and inequalities. In our previous article, we evaluated the expression $ x y - x \left( x - y^0 \right) $ using the order of operations (PEMDAS). In this article, we will answer some frequently asked questions about evaluating algebraic expressions.

Q&A

Q: What is the order of operations (PEMDAS)?

A: The order of operations (PEMDAS) is a set of rules that helps us evaluate mathematical expressions. The acronym PEMDAS stands for:

Parentheses
Exponents
Multiplication and Division
Addition and Subtraction

Q: How do I evaluate an expression with parentheses?

Q: What is the difference between $ y^0 $ and $ y^1 $?

A: The difference between $ y^0 $ and $ y^1 $ is that $ y^0 $ is equal to 1, while $ y^1 $ is equal to $ y $. For example, if $ y = 2 $, then $ y^0 = 1 $ and $ y^1 = 2 $.

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

A: To simplify an expression with multiple terms, we need to combine like terms. Like terms are terms that have the same variable raised to the same power. For example, $ 2x + 3x $ can be simplified to $ 5x $.

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

A: A variable is a letter or symbol that represents a value that can change. A constant is a value that does not change. For example, $ x $ is a variable, while $ 5 $ is a constant.

Q: How do I evaluate an expression with fractions?

A: To evaluate an expression with fractions, we need to follow the order of operations (PEMDAS). First, evaluate any exponential expressions, then evaluate any multiplication and division operations from left to right.

Q: What is the difference between a rational expression and an irrational expression?

A: A rational expression is an expression that can be written as a fraction of two polynomials. An irrational expression is an expression that cannot be written as a fraction of two polynomials.

Q: How do I simplify a rational expression?

A: To simplify a rational expression, we need to factor the numerator and denominator, then cancel out any common factors.

Examples

Example 1: Evaluating an Expression with Parentheses

Evaluate the expression $ (2x + 3) - (x - 2) $.

Solution:

Evaluate the expressions inside the parentheses: $ 2x + 3 $ and $ x - 2 $
Substitute the expressions back into the original expression: $ (2x + 3) - (x - 2) $
Distribute the negative sign: $ 2x + 3 - x + 2 $
Combine like terms: $ x + 5 $

Example 2: Simplifying an Expression with Multiple Terms

Simplify the expression $ 2x + 3x + 4x $.

Solution:

Combine like terms: $ 2x + 3x + 4x = 9x $

Example 3: Evaluating an Expression with Fractions

Evaluate the expression $ \frac{2x}{3} + \frac{3x}{4} $.

Solution:

Find a common denominator: $ 12 $
Rewrite the fractions with the common denominator: $ \frac{8x}{12} + \frac{9x}{12} $
Add the fractions: $ \frac{17x}{12} $

Conclusion

In this article, we answered some frequently asked questions about evaluating algebraic expressions. We covered topics such as the order of operations (PEMDAS), evaluating expressions with parentheses, simplifying expressions with multiple terms, and evaluating expressions with fractions. We also provided some examples to illustrate the concepts.

Additional Resources

If you want to learn more about evaluating algebraic expressions, here are some additional resources:

Khan Academy: Algebraic Expressions
Mathway: Algebraic Expressions
Wolfram Alpha: Algebraic Expressions

Frequently Asked Questions

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

A: A variable is a letter or symbol that represents a value that can change. A constant is a value that does not change. For example, $ x $ is a variable, while $ 5 $ is a constant.

Q: How do I evaluate an expression with fractions?

Q: What is the difference between a rational expression and an irrational expression?

A: A rational expression is an expression that can be written as a fraction of two polynomials. An irrational expression is an expression that cannot be written as a fraction of two polynomials.

Q: How do I simplify a rational expression?

A: To simplify a rational expression, we need to factor the numerator and denominator, then cancel out any common factors.

Conclusion

In this article, we provided some additional resources and answered some frequently asked questions about evaluating algebraic expressions. We hope this article has been helpful in understanding the concepts of evaluating algebraic expressions.