Mt. 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 Of P − 5 Q 0 \frac{p^{-5}}{q^0} Q 0 P − 5 ​ .Four Students Wrote These Expressions:$[ \begin{tabular}{lccc} Isaac & Rosa & Bruce & Brianna

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Understanding the Problem

Mt. Walden presented his students with the expression p−5q0\frac{p^{-5}}{q^0} and asked them to write an equivalent expression. This problem requires an understanding of exponents, specifically negative exponents and zero exponents. In this article, we will explore the concept of equivalent expressions, simplify the given expression, and evaluate the responses of four students: Isaac, Rosa, Bruce, and Brianna.

Negative Exponents

A negative exponent is a fraction with a negative power. It can be rewritten as a positive exponent by taking the reciprocal of the base and changing the sign of the exponent. For example, a−n=1ana^{-n} = \frac{1}{a^n}. This concept is crucial in simplifying the given expression.

Zero Exponents

A zero exponent is a fraction with a power of zero. It can be rewritten as 1, as any non-zero number raised to the power of zero is equal to 1. For example, a0=1a^0 = 1. This concept is also essential in simplifying the given expression.

Simplifying the Expression

The given expression is p−5q0\frac{p^{-5}}{q^0}. To simplify this expression, we can use the concept of negative exponents and zero exponents. We can rewrite the negative exponent as a positive exponent by taking the reciprocal of the base and changing the sign of the exponent. We can also rewrite the zero exponent as 1.

p−5q0=1p51=1p5\frac{p^{-5}}{q^0} = \frac{\frac{1}{p^5}}{1} = \frac{1}{p^5}

Evaluating Student Responses

Now that we have simplified the expression, let's evaluate the responses of the four students.

Isaac

Isaac wrote the expression 1p5\frac{1}{p^5}. This expression is equivalent to the simplified expression we obtained earlier.

Rosa

Rosa wrote the expression p51\frac{p^5}{1}. This expression is not equivalent to the simplified expression we obtained earlier. Rosa incorrectly applied the concept of negative exponents.

Bruce

Bruce wrote the expression 1p−5\frac{1}{p^{-5}}. This expression is equivalent to the simplified expression we obtained earlier. Bruce correctly applied the concept of negative exponents.

Brianna

Brianna wrote the expression p5q5\frac{p^5}{q^5}. This expression is not equivalent to the simplified expression we obtained earlier. Brianna incorrectly applied the concept of zero exponents.

Conclusion

In conclusion, the equivalent expression of p−5q0\frac{p^{-5}}{q^0} is 1p5\frac{1}{p^5}. We evaluated the responses of four students and found that only Isaac and Bruce correctly simplified the expression. Rosa and Brianna incorrectly applied the concept of negative exponents and zero exponents, respectively.

Key Takeaways

  • Negative exponents can be rewritten as positive exponents by taking the reciprocal of the base and changing the sign of the exponent.
  • Zero exponents can be rewritten as 1.
  • Equivalent expressions can be obtained by simplifying the given expression using the concept of negative exponents and zero exponents.

Practice Problems

  1. Simplify the expression a−2b0\frac{a^{-2}}{b^0}.
  2. Simplify the expression c0d3\frac{c^0}{d^3}.
  3. Simplify the expression e−4f0\frac{e^{-4}}{f^0}.

Answer Key

  1. 1a2\frac{1}{a^2}
  2. 1
  3. 1e4\frac{1}{e^4}
    Q&A: Equivalent Expressions and Exponents =============================================

Frequently Asked Questions

In this article, we will address some frequently asked questions related to equivalent expressions and exponents.

Q: What is an equivalent expression?

A: An equivalent expression is an expression that has the same value as another expression. In other words, two expressions are equivalent if they can be simplified to the same value.

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 positive exponent by taking the reciprocal of the base and changing the sign of the exponent. For example, a−n=1ana^{-n} = \frac{1}{a^n}.

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

A: To simplify an expression with a zero exponent, you can rewrite it as 1, as any non-zero number raised to the power of zero is equal to 1. For example, a0=1a^0 = 1.

Q: What is the difference between a negative exponent and a zero exponent?

A: A negative exponent is a fraction with a negative power, while a zero exponent is a fraction with a power of zero. Negative exponents can be rewritten as positive exponents, while zero exponents can be rewritten as 1.

Q: Can you give an example of an expression with a negative exponent?

A: Yes, an example of an expression with a negative exponent is 1a5\frac{1}{a^5}. This expression can be rewritten as a−5a^{-5}.

Q: Can you give an example of an expression with a zero exponent?

A: Yes, an example of an expression with a zero exponent is 1a0\frac{1}{a^0}. This expression can be rewritten as 1.

Q: How do I evaluate an expression with a negative exponent and a zero exponent?

A: To evaluate an expression with a negative exponent and a zero exponent, you can simplify the expression by rewriting the negative exponent as a positive exponent and the zero exponent as 1. For example, a−5a0=1a5\frac{a^{-5}}{a^0} = \frac{1}{a^5}.

Q: Can you give an example of an expression with a negative exponent and a zero exponent?

A: Yes, an example of an expression with a negative exponent and a zero exponent is a−5a0\frac{a^{-5}}{a^0}. This expression can be rewritten as 1a5\frac{1}{a^5}.

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

A: To determine if two expressions are equivalent, you can simplify both expressions and compare their values. If the values are the same, then the expressions are equivalent.

Q: Can you give an example of two equivalent expressions?

A: Yes, an example of two equivalent expressions is a−5a0\frac{a^{-5}}{a^0} and 1a5\frac{1}{a^5}. Both expressions have the same value, so they are equivalent.

Conclusion

In conclusion, equivalent expressions and exponents are important concepts in mathematics. By understanding how to simplify expressions with negative exponents and zero exponents, you can evaluate expressions and determine if they are equivalent. We hope this Q&A article has helped you understand these concepts better.

Practice Problems

  1. Simplify the expression a−2b0\frac{a^{-2}}{b^0}.
  2. Simplify the expression c0d3\frac{c^0}{d^3}.
  3. Simplify the expression e−4f0\frac{e^{-4}}{f^0}.

Answer Key

  1. 1a2\frac{1}{a^2}
  2. 1
  3. 1e4\frac{1}{e^4}