Which Expression Is Equivalent To $5 Y^{-3}$?A. $\frac{1}{125 Y^3}$ B. $ 1 5 Y 3 \frac{1}{5 Y^3} 5 Y 3 1 ​ [/tex] C. $\frac{5}{y^3}$ D. $\frac{125}{y^3}$

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Understanding Exponential Expressions

Exponential expressions are a fundamental concept in mathematics, used to describe the relationship between variables and their exponents. In this article, we will explore the concept of equivalent forms of exponential expressions, focusing on the expression $5 y^{-3}$. We will examine the given options and determine which one is equivalent to the given expression.

The Concept of Negative Exponents

Before we dive into the solution, it's essential to understand the concept of negative exponents. A negative exponent indicates that the base is being raised to a power that is the reciprocal of the given exponent. In other words, $y^{-3}$ is equivalent to $\frac{1}{y^3}$.

Analyzing the Given Options

Now that we have a solid understanding of negative exponents, let's analyze the given options:

A. $\frac{1}{125 y^3}$ B. $\frac{1}{5 y^3}$ C. $\frac{5}{y^3}$ D. $\frac{125}{y^3}$

Option A: $\frac{1}{125 y^3}$

To determine if option A is equivalent to $5 y^{-3}$, we need to simplify the expression. We can start by rewriting the negative exponent as a positive exponent:

5y−3=5⋅1y35 y^{-3} = 5 \cdot \frac{1}{y^3}

Now, we can simplify the expression by multiplying the numerator and denominator:

5⋅1y3=5y35 \cdot \frac{1}{y^3} = \frac{5}{y^3}

However, this is not option A. Option A has a denominator of $125 y^3$, which is not equivalent to $y^3$.

Option B: $\frac{1}{5 y^3}$

Let's analyze option B. We can rewrite the negative exponent as a positive exponent:

5y−3=5⋅1y35 y^{-3} = 5 \cdot \frac{1}{y^3}

Now, we can simplify the expression by multiplying the numerator and denominator:

5⋅1y3=5y35 \cdot \frac{1}{y^3} = \frac{5}{y^3}

However, this is not option B. Option B has a denominator of $5 y^3$, which is not equivalent to $y^3$.

Option C: $\frac{5}{y^3}$

Let's analyze option C. We can rewrite the negative exponent as a positive exponent:

5y−3=5⋅1y35 y^{-3} = 5 \cdot \frac{1}{y^3}

Now, we can simplify the expression by multiplying the numerator and denominator:

5⋅1y3=5y35 \cdot \frac{1}{y^3} = \frac{5}{y^3}

This is option C! We have found an equivalent form of the given expression.

Option D: $\frac{125}{y^3}$

Let's analyze option D. We can rewrite the negative exponent as a positive exponent:

5y−3=5⋅1y35 y^{-3} = 5 \cdot \frac{1}{y^3}

Now, we can simplify the expression by multiplying the numerator and denominator:

5⋅1y3=5y35 \cdot \frac{1}{y^3} = \frac{5}{y^3}

However, this is not option D. Option D has a numerator of $125$, which is not equivalent to $5$.

Conclusion

In conclusion, the equivalent form of $5 y^-3}$ is option C $\frac{5{y^3}$. We were able to simplify the expression by rewriting the negative exponent as a positive exponent and multiplying the numerator and denominator.

Key Takeaways

  • Negative exponents indicate that the base is being raised to a power that is the reciprocal of the given exponent.
  • To simplify an exponential expression with a negative exponent, rewrite the negative exponent as a positive exponent and multiply the numerator and denominator.
  • The equivalent form of $5 y^-3}$ is option C $\frac{5{y^3}$.

Practice Problems

  1. Simplify the expression $2 x^{-2}$.
  2. Simplify the expression $3 y^{-4}$.
  3. Simplify the expression $4 z^{-1}$.

Answer Key

  1. 2x2\frac{2}{x^2}

  2. 3y4\frac{3}{y^4}

  3. \frac{4}{z}$<br/>

Q: What is the rule for simplifying exponential expressions with negative exponents?

A: To simplify an exponential expression with a negative exponent, rewrite the negative exponent as a positive exponent and multiply the numerator and denominator.

Q: How do I rewrite a negative exponent as a positive exponent?

A: To rewrite a negative exponent as a positive exponent, simply change the sign of the exponent. For example, $y^{-3}$ becomes $\frac{1}{y^3}$.

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

A: A negative exponent indicates that the base is being raised to a power that is the reciprocal of the given exponent. A positive exponent indicates that the base is being raised to a power that is the same as the given exponent.

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

A: Let's simplify the expression $5 y^{-3}$. We can rewrite the negative exponent as a positive exponent:

5y−3=5⋅1y35 y^{-3} = 5 \cdot \frac{1}{y^3}

Now, we can simplify the expression by multiplying the numerator and denominator:

5⋅1y3=5y35 \cdot \frac{1}{y^3} = \frac{5}{y^3}

Q: What is the equivalent form of $5 y^{-3}$?

A: The equivalent form of $5 y^{-3}$ is $\frac{5}{y^3}$.

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

A: Let's simplify the expression $2 x^{-2}$. We can rewrite the negative exponent as a positive exponent:

2x−2=2⋅1x22 x^{-2} = 2 \cdot \frac{1}{x^2}

Now, we can simplify the expression by multiplying the numerator and denominator:

2⋅1x2=2x22 \cdot \frac{1}{x^2} = \frac{2}{x^2}

Q: What is the equivalent form of $2 x^{-2}$?

A: The equivalent form of $2 x^{-2}$ is $\frac{2}{x^2}$.

Q: Can you give a practice problem for simplifying exponential expressions with negative exponents?

A: Simplify the expression $3 y^{-4}$.

Answer: $\frac{3}{y^4}$

Q: Can you give a practice problem for simplifying exponential expressions with negative exponents?

A: Simplify the expression $4 z^{-1}$.

Answer: $\frac{4}{z}$

Q: What are some common mistakes to avoid when simplifying exponential expressions with negative exponents?

A: Some common mistakes to avoid when simplifying exponential expressions with negative exponents include:

  • Not rewriting the negative exponent as a positive exponent
  • Not multiplying the numerator and denominator
  • Not simplifying the expression correctly

Q: How can I practice simplifying exponential expressions with negative exponents?

A: You can practice simplifying exponential expressions with negative exponents by working through practice problems, such as the ones listed above. You can also try creating your own practice problems to challenge yourself.

Conclusion

In conclusion, simplifying exponential expressions with negative exponents is a straightforward process that involves rewriting the negative exponent as a positive exponent and multiplying the numerator and denominator. By following these steps and practicing regularly, you can become proficient in simplifying exponential expressions with negative exponents.