Simplify. Express Your Answer As A Single Term, Without A Denominator.$u^6 V W^{-3} \cdot U^0 V^2 W^{-9}$

Feb 28, 2025 by ADMIN 106 views

**Simplify Expressions: A Comprehensive Guide**

Introduction

In mathematics, simplifying expressions is a crucial skill that helps us solve problems efficiently and accurately. When dealing with algebraic expressions, we often encounter terms with exponents, which can be simplified using the rules of exponentiation. In this article, we will explore the process of simplifying expressions with exponents, focusing on the given expression: $u^6 v w^{-3} \cdot u^0 v^2 w^{-9}$ .

Understanding Exponents

Before we dive into simplifying the given expression, let's review the basics of exponents. An exponent is a small number that is raised to the power of a variable or a constant. For example, in the expression $u^6$ , the exponent 6 is raised to the power of the variable $u$ . When we multiply two or more terms with the same base, we add their exponents. For instance, $u^6 \cdot u^3 = u^{6+3} = u^9$ .

Simplifying the Given Expression

Now, let's apply the rules of exponentiation to simplify the given expression: $u^6 v w^{-3} \cdot u^0 v^2 w^{-9}$ . To simplify this expression, we need to follow the order of operations (PEMDAS):

Multiply the coefficients: When multiplying two or more terms, we multiply their coefficients. In this case, the coefficients are $u^6$ , $v$ , and $w^{-3}$ , and $u^0$ , $v^2$ , and $w^{-9}$ .
Add the exponents: When multiplying two or more terms with the same base, we add their exponents. In this case, we have two terms with the base $u$ , two terms with the base $v$ , and two terms with the base $w$ .
Simplify the resulting expression: After adding the exponents, we simplify the resulting expression by combining like terms.

Step 1: Multiply the Coefficients

When multiplying two or more terms, we multiply their coefficients. In this case, we have:

$u^6 v w^{-3} \cdot u^0 v^2 w^{-9} = (u^6 \cdot u^0) \cdot (v \cdot v^2) \cdot (w^{-3} \cdot w^{-9})$

Step 2: Add the Exponents

When multiplying two or more terms with the same base, we add their exponents. In this case, we have:

$(u^6 \cdot u^0) = u^{6+0} = u^6$

$(v \cdot v^2) = v^{1+2} = v^3$

$(w^{-3} \cdot w^{-9}) = w^{-3-9} = w^{-12}$

Step 3: Simplify the Resulting Expression

After adding the exponents, we simplify the resulting expression by combining like terms:

$u^6 v^3 w^{-12}$

Conclusion

In this article, we simplified the expression $u^6 v w^{-3} \cdot u^0 v^2 w^{-9}$ using the rules of exponentiation. We followed the order of operations (PEMDAS) and applied the rules of multiplying coefficients and adding exponents. The resulting simplified expression is $u^6 v^3 w^{-12}$ . This example demonstrates the importance of simplifying expressions with exponents, which is a crucial skill in mathematics.

Common Mistakes to Avoid

When simplifying expressions with exponents, it's essential to avoid common mistakes. Here are a few:

Not following the order of operations: Make sure to follow the order of operations (PEMDAS) when simplifying expressions with exponents.
Not adding exponents correctly: When multiplying two or more terms with the same base, add their exponents correctly.
Not combining like terms: After adding exponents, simplify the resulting expression by combining like terms.

Real-World Applications

Simplifying expressions with exponents has numerous real-world applications. Here are a few:

Science and Engineering: Exponents are used to describe the behavior of physical systems, such as population growth, chemical reactions, and electrical circuits.
Finance: Exponents are used to calculate interest rates, investment returns, and financial derivatives.
Computer Science: Exponents are used to describe the complexity of algorithms, data structures, and computational models.

Final Thoughts

Q&A: Simplifying Expressions with Exponents

In this article, we will answer some frequently asked questions about simplifying expressions with exponents.

Q: What is the order of operations when simplifying expressions with exponents?

A: The order of operations when simplifying expressions with exponents 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.

Q: How do I add exponents when multiplying two or more terms with the same base?

A: When multiplying two or more terms with the same base, you add their exponents. For example:

$u^6 \cdot u^3 = u^{6+3} = u^9$

Q: What is the rule for multiplying two or more terms with different bases?

A: When multiplying two or more terms with different bases, you multiply their coefficients and add their exponents. For example:

$u^6 v w^{-3} \cdot u^0 v^2 w^{-9} = (u^6 \cdot u^0) \cdot (v \cdot v^2) \cdot (w^{-3} \cdot w^{-9})$

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

A: When simplifying an expression with a negative exponent, you can rewrite it as a fraction with a positive exponent. For example:

$u^{-3} = \frac{1}{u^3}$

Q: What is the rule for dividing two or more terms with the same base?

A: When dividing two or more terms with the same base, you subtract their exponents. For example:

$\frac{u^6}{u^3} = u^{6-3} = u^3$

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

A: When simplifying an expression with a zero exponent, you can rewrite it as 1. For example:

$u^0 = 1$

Q: What is the rule for simplifying an expression with a variable in the exponent?

A: When simplifying an expression with a variable in the exponent, you can rewrite it as a product of the variable and the exponent. For example:

$u^{2x} = u^2 \cdot u^x$

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

A: When simplifying an expression with a fraction in the exponent, you can rewrite it as a product of the fraction and the exponent. For example:

$u^{\frac{1}{2}} = \sqrt{u}$

Conclusion

In this article, we answered some frequently asked questions about simplifying expressions with exponents. We hope this Q&A guide has provided a comprehensive overview of the rules and techniques for simplifying expressions with exponents.

Common Mistakes to Avoid

When simplifying expressions with exponents, it's essential to avoid common mistakes. Here are a few:

Not following the order of operations: Make sure to follow the order of operations (PEMDAS) when simplifying expressions with exponents.
Not adding exponents correctly: When multiplying two or more terms with the same base, add their exponents correctly.
Not combining like terms: After adding exponents, simplify the resulting expression by combining like terms.

Real-World Applications

Simplifying expressions with exponents has numerous real-world applications. Here are a few:

Science and Engineering: Exponents are used to describe the behavior of physical systems, such as population growth, chemical reactions, and electrical circuits.
Finance: Exponents are used to calculate interest rates, investment returns, and financial derivatives.
Computer Science: Exponents are used to describe the complexity of algorithms, data structures, and computational models.

Final Thoughts

Simplifying expressions with exponents is a fundamental skill in mathematics that has numerous real-world applications. By following the rules of exponentiation and avoiding common mistakes, we can simplify complex expressions and solve problems efficiently and accurately. In this article, we provided a comprehensive guide to simplifying expressions with exponents, including a Q&A section and real-world applications.