Mr. Walden Wrote The Expression P − 5 Q 0 \frac{p^{-5}}{q^0} Q 0 P − 5 . He Asked His Students To Write An Equivalent Expression.Four Students Wrote These Expressions:- Rosa: P − 5 + Q 0 P^{-5} + Q^0 P − 5 + Q 0 - Bruce: P 5 Q 0 P^5 Q^0 P 5 Q 0 - Brianna: $\frac{1}{p^5

Mar 13, 2025 by ADMIN 272 views

**Understanding Equivalent Expressions in Algebra**

In algebra, equivalent expressions are mathematical expressions that have the same value or result, even if they are written differently. Mr. Walden's students were asked to write an equivalent expression to the given expression $\frac{p$ . In this article, we will analyze the expressions written by four students: Rosa, Bruce, Brianna, and Henry, and determine which one is equivalent to the given expression.

The Given Expression

The given expression is $\frac{p^{-5}}{q^0}$ . To understand this expression, we need to recall the rules of exponents. When a variable is raised to a negative power, it is equivalent to taking the reciprocal of the variable raised to the positive power. In other words, $p^{-5}$ is equivalent to $\frac{1}{p^5}$ . Additionally, any number raised to the power of 0 is equal to 1.

Rosa's Expression

Rosa wrote the expression $p^{-5} + q^0$ . However, this expression is not equivalent to the given expression. The reason is that the given expression is a fraction, while Rosa's expression is an addition of two terms. To make Rosa's expression equivalent to the given expression, we need to rewrite it as a fraction.

p^{-5} + q^0 = \frac{p^{-5}}{1} + \frac{q^0}{1} = \frac{p^{-5} + q^0}{1}

However, this expression is still not equivalent to the given expression. The reason is that the denominator of the fraction is 1, while the given expression has a denominator of $q^0$ . To make Rosa's expression equivalent to the given expression, we need to rewrite it with a denominator of $q^0$ .

p^{-5} + q^0 = \frac{p^{-5} + q^0}{q^0}

However, this expression is still not equivalent to the given expression. The reason is that the numerator of the fraction is $p^{-5} + q^0$ , while the given expression has a numerator of $p^{-5}$ . To make Rosa's expression equivalent to the given expression, we need to rewrite it with a numerator of $p^{-5}$ .

p^{-5} + q^0 = \frac{p^{-5}}{q^0} + \frac{q^0}{q^0} = \frac{p^{-5} + q^0}{q^0}

p^{-5} + q^0 = \frac{p^{-5}}{q^0} + \frac{q^0}{q^0} = \frac{p^{-5}}{q^0} + 1

However, this expression is still not equivalent to the given expression. The reason is that the expression has an additional term of 1, while the given expression does not have any additional terms.

Bruce's Expression

Bruce wrote the expression $p^5 q^0$ . However, this expression is not equivalent to the given expression. The reason is that the given expression has a negative exponent, while Bruce's expression has a positive exponent. To make Bruce's expression equivalent to the given expression, we need to rewrite it with a negative exponent.

p^5 q^0 = \frac{p^5}{q^0}

However, this expression is still not equivalent to the given expression. The reason is that the numerator of the fraction is $p^5$ , while the given expression has a numerator of $p^{-5}$ . To make Bruce's expression equivalent to the given expression, we need to rewrite it with a numerator of $p^{-5}$ .

p^5 q^0 = \frac{p^{-5}}{q^0}

However, this expression is equivalent to the given expression.

Brianna's Expression

Brianna wrote the expression $\frac{1}{p^5}$ . However, this expression is not equivalent to the given expression. The reason is that the given expression has a denominator of $q^0$ , while Brianna's expression has a denominator of $p^5$ . To make Brianna's expression equivalent to the given expression, we need to rewrite it with a denominator of $q^0$ .

\frac{1}{p^5} = \frac{1}{p^5} \cdot \frac{q^0}{q^0} = \frac{q^0}{p^5 q^0} = \frac{q^0}{p^5 q^0} \cdot \frac{p^5}{p^5} = \frac{q^0 p^5}{p^5 q^0 p^5} = \frac{q^0 p^5}{p^{10} q^0}

However, this expression is still not equivalent to the given expression. The reason is that the numerator of the fraction is $q^0 p^5$ , while the given expression has a numerator of $p^{-5}$ . To make Brianna's expression equivalent to the given expression, we need to rewrite it with a numerator of $p^{-5}$ .

\frac{1}{p^5} = \frac{p^{-5}}{q^0}

However, this expression is equivalent to the given expression.

Henry's Expression

Henry wrote the expression $\frac{p^{-5}}{q^0}$ . This expression is equivalent to the given expression.

Conclusion

In conclusion, the equivalent expression to the given expression $\frac{p^{-5}}{q^0}$ is $\frac{p^{-5}}{q^0}$ . The expressions written by Rosa, Bruce, and Brianna are not equivalent to the given expression, while Henry's expression is equivalent to the given expression.

Key Takeaways

Equivalent expressions are mathematical expressions that have the same value or result, even if they are written differently.
To determine if two expressions are equivalent, we need to simplify them and compare their values.
The given expression $\frac{p^{-5}}{q^0}$ is equivalent to the expression $\frac{p^{-5}}{q^0}$ .
The expressions written by Rosa, Bruce, and Brianna are not equivalent to the given expression, while Henry's expression is equivalent to the given expression.

Final Answer

In our previous article, we discussed equivalent expressions in algebra and analyzed the expressions written by four students: Rosa, Bruce, Brianna, and Henry. In this article, we will answer some frequently asked questions about equivalent expressions in algebra.

Q: What is an equivalent expression?

A: An equivalent expression is a mathematical expression that has the same value or result as another expression, even if they are written differently.

Q: How do I determine if two expressions are equivalent?

A: To determine if two expressions are equivalent, you need to simplify them and compare their values. You can use algebraic properties and rules to simplify the expressions and make them easier to compare.

Q: What are some common algebraic properties and rules that I can use to simplify expressions?

A: Some common algebraic properties and rules that you can use to simplify expressions include:

The commutative property of addition and multiplication
The associative property of addition and multiplication
The distributive property of multiplication over addition
The rule for multiplying and dividing exponents with the same base
The rule for adding and subtracting fractions with the same denominator

Q: How do I simplify expressions using algebraic properties and rules?

A: To simplify expressions using algebraic properties and rules, you need to apply the properties and rules in the correct order. Here are some steps you can follow:

Simplify any expressions inside parentheses or brackets.
Apply the commutative property of addition and multiplication.
Apply the associative property of addition and multiplication.
Apply the distributive property of multiplication over addition.
Simplify any expressions with exponents.
Simplify any fractions.

Q: What are some common mistakes to avoid when simplifying expressions?

A: Some common mistakes to avoid when simplifying expressions include:

Forgetting to simplify expressions inside parentheses or brackets.
Applying the commutative property of addition and multiplication in the wrong order.
Forgetting to simplify expressions with exponents.
Forgetting to simplify fractions.

Q: How do I know if an expression is equivalent to another expression?

A: To know if an expression is equivalent to another expression, you need to compare their values. You can use algebraic properties and rules to simplify the expressions and make them easier to compare.

Q: What are some real-world applications of equivalent expressions in algebra?

A: Equivalent expressions in algebra have many real-world applications, including:

Simplifying complex mathematical expressions in physics and engineering.
Solving systems of linear equations in economics and finance.
Modeling population growth and decay in biology and ecology.
Analyzing data and making predictions in statistics and data science.

Q: How can I practice simplifying expressions and determining equivalent expressions?

A: You can practice simplifying expressions and determining equivalent expressions by:

Working through algebraic exercises and problems.
Using online resources and tools, such as algebraic calculators and software.
Joining online communities and forums to discuss algebra and get help from others.
Taking online courses or tutorials to learn more about algebra and equivalent expressions.

Conclusion

In conclusion, equivalent expressions in algebra are an important concept that has many real-world applications. By understanding how to simplify expressions and determine equivalent expressions, you can solve complex mathematical problems and make predictions in various fields. Remember to practice simplifying expressions and determining equivalent expressions to become proficient in algebra.

Key Takeaways

Equivalent expressions are mathematical expressions that have the same value or result as another expression, even if they are written differently.
To determine if two expressions are equivalent, you need to simplify them and compare their values.
Algebraic properties and rules, such as the commutative property of addition and multiplication, can be used to simplify expressions.
Real-world applications of equivalent expressions in algebra include simplifying complex mathematical expressions, solving systems of linear equations, and modeling population growth and decay.

Final Answer

The final answer is $\boxed{\frac{p^{-5}}{q^0}}$ .