Rewrite The Expression In Simplest Form. Be Sure To Express The Answer Using Positive Exponents.$\[ (11 X^7 Y) \cdot (2 X^2 Y^5) \\]Enter Your Answer In The Box Provided.

Feb 28, 2025 by ADMIN 171 views

Understanding the Problem

When dealing with algebraic expressions, it's essential to simplify them to their most basic form. This involves combining like terms, eliminating any unnecessary components, and expressing the result using positive exponents. In this article, we'll focus on rewriting the given expression in its simplest form, ensuring that the answer is presented using positive exponents.

The Given Expression

The expression we need to simplify is:

{ (11 x^7 y) \cdot (2 x^2 y^5) \}

Breaking Down the Expression

To simplify this expression, we need to apply the rules of exponents. When multiplying two terms with the same base, we add their exponents. In this case, both terms have the base $x$ and $y$ . We'll start by multiplying the coefficients (the numbers in front of the variables) and then add the exponents of the variables.

Multiplying the Coefficients

The coefficients of the two terms are 11 and 2. When multiplying these numbers, we get:

$11 \cdot 2 = 22$

Adding the Exponents of the Variables

Now, let's focus on the variables. We have $x^7$ and $x^2$ , and $y$ and $y^5$ . When multiplying these terms, we add their exponents. For the variable $x$ , we get:

$7 + 2 = 9$

For the variable $y$ , we get:

$1 + 5 = 6$

Combining the Results

Now that we've multiplied the coefficients and added the exponents of the variables, we can combine the results to get the simplified expression:

$22x^9y^6$

Expressing the Answer Using Positive Exponents

The simplified expression is already in its simplest form, and it's expressed using positive exponents. The variable $x$ has an exponent of 9, and the variable $y$ has an exponent of 6.

Conclusion

In this article, we've rewritten the given expression in its simplest form, ensuring that the answer is presented using positive exponents. By applying the rules of exponents and combining like terms, we've arrived at the final expression: $22x^9y^6$ . This result demonstrates the importance of simplifying algebraic expressions to their most basic form, which is essential in various mathematical applications.

Frequently Asked Questions

What is the simplified form of the expression $(11 x^7 y) \cdot (2 x^2 y^5)$ $(11 x^{7} y) \cdot (2 x^{2} y^{5})$ ?
- The simplified form of the expression is $22x^9y^6$ .
How do you add exponents when multiplying two terms with the same base?
- When multiplying two terms with the same base, you add their exponents.
What is the rule for multiplying coefficients?
- When multiplying coefficients, you multiply the numbers together.

Additional Resources

Step-by-Step Solution

Multiply the coefficients: $11 \cdot 2 = 22$
Add the exponents of the variables: $7 + 2 = 9$ for $x$ , and $1 + 5 = 6$ for $y$
Combine the results: $22x^9y^6$

Final Answer

The final answer is: $\boxed{22x^9y^6}$

Understanding Algebraic Expressions

Algebraic expressions are a fundamental concept in mathematics, and simplifying them is a crucial skill to master. In this article, we'll address some common questions and provide detailed explanations to help you better understand how to simplify algebraic expressions.

Q1: What is an Algebraic Expression?

A1: An algebraic expression is a mathematical expression that consists of variables, constants, and mathematical operations. It's a way to represent a value or a relationship between values using letters, numbers, and symbols.

Q2: How Do I Simplify an Algebraic Expression?

A2: To simplify an algebraic expression, you need to combine like terms, eliminate any unnecessary components, and express the result using positive exponents. This involves applying the rules of exponents, such as multiplying coefficients and adding exponents of variables.

Q3: What is a Like Term?

A3: A like term is a term that has the same variable(s) raised to the same power. For example, $x^2$ and $x^2$ are like terms, while $x^2$ and $x^3$ are not.

Q4: How Do I Combine Like Terms?

A4: To combine like terms, you add or subtract the coefficients of the like terms. For example, if you have $x^2 + 3x^2$ , you can combine the like terms by adding the coefficients: $x^2 + 3x^2 = 4x^2$ .

Q5: What is the Rule for Multiplying Exponents?

A5: When multiplying two terms with the same base, you add their exponents. For example, if you have $x^2 \cdot x^3$ , you can multiply the exponents by adding them: $x^2 \cdot x^3 = x^{2+3} = x^5$ .

Q6: How Do I Simplify an Expression with Negative Exponents?

A6: To simplify an expression with negative exponents, you can rewrite the expression using positive exponents. For example, if you have $\frac{1}{x^2}$ , you can rewrite it as $x^{-2}$ , and then simplify it by multiplying the exponent by -1: $x^{-2} = \frac{1}{x^2}$ .

Q7: What is the Order of Operations?

A7: The order of operations is a set of rules that dictates the order in which you perform mathematical operations. The order of operations is:

Parentheses: Evaluate expressions inside parentheses first.
Exponents: Evaluate any exponential expressions next.
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.

Q8: How Do I Simplify an Expression with Multiple Variables?

A8: To simplify an expression with multiple variables, you need to apply the rules of exponents and combine like terms. For example, if you have $(x^2y^3) \cdot (x^4y^2)$ , you can simplify it by multiplying the exponents of the variables and combining like terms: $(x^2y^3) \cdot (x^4y^2) = x^{2+4}y^{3+2} = x^6y^5$ .

Q9: What is the Difference Between a Variable and a Constant?

A9: 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.

Q10: How Do I Simplify an Expression with Fractions?

A10: To simplify an expression with fractions, you need to apply the rules of exponents and combine like terms. For example, if you have $\frac{x^2}{y^3} \cdot \frac{y^2}{x^4}$ , you can simplify it by multiplying the exponents of the variables and combining like terms: $\frac{x^2}{y^3} \cdot \frac{y^2}{x^4} = \frac{x^2y^2}{y^3x^4} = \frac{y^2}{x^2y^3}$ .