Using the properties of exponents to generate an equivalent expression, which of the following would be the rational exponent on an expression equivalent to 216−−−√3 ?(1 point)

Responses

13
Start Fraction 1 over 3 End Fraction

−13
negative Start Fraction 1 over 3 End Fraction

−3
negative 3

3
3

13 answers

To express the expression \( 216^{-\frac{1}{3}} \) using rational exponents, we can follow these steps:

  1. Recognize that \( 216^{-\frac{1}{3}} \) can be rewritten as \( \frac{1}{216^{\frac{1}{3}}} \). This means we are looking for the cube root of 216 and taking its reciprocal.
  2. The expression \( 216^{\frac{1}{3}} \) refers to the cube root of 216, which is 6 because \( 6^3 = 216 \).
  3. Therefore, \( 216^{-\frac{1}{3}} = \frac{1}{6} \).

However, the question directly asks for the rational exponent on the expression equivalent to \( 216^{-\frac{1}{3}} \).

Since the expression \( 216^{-\frac{1}{3}} \) has a base of 216 raised to the power of \(-\frac{1}{3}\), the rational exponent on this expression would simply be:

Response: \( -\frac{1}{3} \)

Thus, the answer is:

−\(\frac{1}{3}\)

Which of the following is an equivalent expression for 512125√3 when simplified using exponent properties?(1 point)
Responses

5√5
Start Fraction Start Root 5 End Root over 5 End Fraction

15
Start Fraction 1 over 5 End Fraction

50
5 superscript 0 baseline

1

To simplify the expression \( 512^{\frac{1}{125}\sqrt{3}} \) using exponent properties, we need to analyze it carefully. However, it looks like the expression might not have been interpreted correctly.

If you meant \( \sqrt[3]{512^{125}} \), we can first rewrite \( 512 \) as a power of \( 2 \): \[ 512 = 2^9 \]

Now substituting for \( 512 \), we have: \[ \sqrt[3]{(2^9)^{125}} = \sqrt[3]{2^{1125}} = 2^{\frac{1125}{3}} = 2^{375} \]

This yields a simple number, but the choices given don't look like they apply directly.

Instead, let's double-check what simplification could lead us to. Based on the options provided:

  1. \( 5\sqrt{5} \)
  2. \( \frac{\sqrt{5}}{5} \)
  3. \( 15 \)
  4. \( \frac{1}{5} \)
  5. \( 50 \)
  6. \( 5^0 \)
  7. \( 1 \)

Since the derived expression \( 2^{375} \) does not simplify to any of these values neatly, let's go back to considering these options as they are written and not as derived from a specific numeral estimation.

Since there's ambiguity here regarding how the expression \( 512125 \) and the radicals connect or if they were meant as separate parts, can you please clarify the arithmetic of \( 512125 \sqrt{3} \)?

If the intention was just \( 512^{1/125} \sqrt{3} \) instead, we'd deal differently. Please confirm or clarify so that the correct manipulation based on that can be explored.

If this was a direct confusion, and if we do stick to \( 512^{\text{something}} \) generally, and if you've worked through, you could also consider something like:

  • \( 5^2 = 25 \)

Given the choices, while more interpretation is needed, you could possibly consider if anything represents a form of \( 25 \) or even decomposing in terms of those options validly.

In absence of clarity, if only one possible based on instinct to option please base on common power roots involved I would suggest \( 5^0 \) = \( 1 \) – since it's an equivalency general rooted towards \( 1 \).

Hopefully, provide clarity or confirm what intent of transformation on \( 512125 \sqrt{3} \) would you like most rightly addressed.

Which of the following is an equivalent expression to 81−12 ?(1 point)
Responses

−181√
negative Start Fraction 1 over Start Root 81 End Root End Fraction

181√
Start Fraction 1 over Start Root 81 End Root End Fraction

81−−√
Start Root 81 End Root

−81−−√
negative Start Root 81 End Root

To simplify the expression \( 81^{-12} \), we can use the properties of exponents.

The negative exponent indicates that we can take the reciprocal of the base raised to the absolute value of the exponent:

\[ 81^{-12} = \frac{1}{81^{12}} \]

However, we can also express \( 81 \) as a power of \( 3 \):

\[ 81 = 3^4 \]

So we can express:

\[ 81^{12} = (3^4)^{12} = 3^{48} \]

Thus:

\[ 81^{-12} = \frac{1}{81^{12}} = \frac{1}{3^{48}} \]

The options given in your question do not seem to relate directly to this computation. Let's re-examine the choices:

  1. \( -\frac{1}{\sqrt{81}} \)
  2. \( \frac{1}{\sqrt{81}} \)
  3. \( \sqrt{81} \)
  4. \( -\sqrt{81} \)

Since \( \sqrt{81} = 9 \), we have:

  • \( -\sqrt{81} = -9 \)
  • \( \sqrt{81} = 9 \)

Now note that \( \frac{1}{\sqrt{81}} = \frac{1}{9} \).

However, \( 81^{-12} \) is not the same as these specific options (in attribution of powers applied).

If we separately analyze from \( 81^{-12} \), the rational simplifications don't reflect an immediate connection back to roots.

Conclusion: If approaching to match against the simplifications, the only equivalent expression directly reducible from straightforward properties based on these responses suggesting denominator factors would be:

Response: \( \frac{1}{\sqrt{81}} \)

Which can be attained before rational concerns arise from reciprocations. Thus, the equivalent and ideal expression, interpreting inversely might be best conveyed contextually as the clean outcome. If these reflect among choices provided, evaluating those forms held, \(\frac{1}{\sqrt{81}}\) stands directly operable.

Use the properties of exponents to determine whether the equation 10−13=−10−−√3 is true or false. Select the response that correctly answers the question and provides the appropriate justification.(1 point)
Responses

False, because according to the Negative Exponent Rule, the equation should be 10−13=−110√3.
False, because according to the Negative Exponent Rule, the equation should be 10 superscript negative Start Fraction 1 over 3 End Fraction baseline equals negative Start Fraction 1 over root index 3 Start Root 10 End Root End Fraction .

False, because the Negative Exponent Rule requires that the number be rewritten as a quotient, not as a negative value.
False, because the Negative Exponent Rule requires that the number be rewritten as a quotient, not as a negative value.

True, because the Negative Exponent Rule requires that the number be rewritten as a negative value.
True, because the Negative Exponent Rule requires that the number be rewritten as a negative value.

True, because the negative sign moves to the front when writing an equivalent expression.

To determine whether the equation \( 10^{-13} = -\sqrt[3]{10} \) is true or false, we can simplify both sides and see if they are equivalent.

  1. Left Side: \[ 10^{-13} = \frac{1}{10^{13}} \quad \text{(using the Negative Exponent Rule)} \]

  2. Right Side: The expression \(-\sqrt[3]{10}\) is equivalent to: \[ -10^{\frac{1}{3}} \quad \text{(which means the negative sign is in front of the cube root of 10)} \]

Now, we need to compare \( \frac{1}{10^{13}} \) and \(-10^{\frac{1}{3}}\).

Clearly, \( \frac{1}{10^{13}} \) is a very small positive number (since \( 10^{13} \) is a very large number), while \(-10^{\frac{1}{3}}\) is a negative number.

Since a positive number cannot equal a negative number, the equation \( 10^{-13} = -\sqrt[3]{10} \) is false.

Conclusion:

The correct response is:

False, because according to the Negative Exponent Rule, the equation should be \( 10^{-13} = \frac{1}{10^{13}} \) and it cannot equal a negative value, \(-\sqrt[3]{10}\).

This response directly relates to the definitions and transformations acknowledged with proper exponent rules and provides a clear justification.

Is the equation 3−5⋅(81√4)33−4=3−6 true or false? Select the response that correctly answers the question and provides the appropriate justification.(1 point)
Responses

True, but the equation should be simplified using the Negative Exponent Rule.
True, but the equation should be simplified using the Negative Exponent Rule.

False. There is an error in calculation using the Quotient Property.
False. There is an error in calculation using the Quotient Property.

False. There is an error in applying the Product Property.
False. There is an error in applying the Product Property.

True. All rules have been applied correctly.

To determine if the equation \( 3^{-5} \cdot (81\sqrt{4})^{3} \cdot 3^{-4} = 3^{-6} \) is true or false, we need to simplify both sides step by step.

Left Side Simplification:

We need to simplify \( 3^{-5} \cdot (81\sqrt{4})^{3} \cdot 3^{-4} \).

  1. Simplifying \( (81\sqrt{4})^{3} \):
    • First, note that \( 81 = 3^4 \) and \( \sqrt{4} = 2 \).
    • Therefore, \( \sqrt{4} = 2 \), so: \[ 81\sqrt{4} = 3^4 \cdot 2. \]
    • Now, we calculate \( (81\sqrt{4})^3 = (3^4 \cdot 2)^3 = (3^4)^3 \cdot 2^3 = 3^{12} \cdot 8 \).

So, we now have: \[ 3^{-5} \cdot (81\sqrt{4})^{3} \cdot 3^{-4} = 3^{-5} \cdot (3^{12} \cdot 8) \cdot 3^{-4}. \]

  1. Combining the exponents of \( 3 \): \[ 3^{-5} \cdot 3^{12} \cdot 3^{-4} = 3^{-5 + 12 - 4} = 3^{3}. \]

Thus, the left side simplifies to: \[ 3^{3} \cdot 8 = 8 \cdot 3^{3}. \]

Right Side:

The right side is \( 3^{-6} \).

Final Comparison:

Now we will compare \( 8 \cdot 3^{3} \) to \( 3^{-6} \):

  • \( 3^{3} = 27 \), so \( 8 \cdot 3^{3} = 8 \cdot 27 = 216 \).
  • \( 3^{-6} = \frac{1}{3^6} = \frac{1}{729} \).

Clearly, \( 216 \) does not equal \( \frac{1}{729} \).

Conclusion:

The equation is false.

Response:

False. There is an error in calculation using the Product Property.

This selection reflects that the conclusion is based on incorrect assumptions or misapplied properties within the steps that led to false equality.

are all the answers correct?

Let's summarize the two previous questions and the answers I provided to ensure clarity and correctness:

  1. First Question: Is \( 10^{-13} = -\sqrt[3]{10} \) true or false?

    • Answer Provided: False, because according to the Negative Exponent Rule, the equation should be \( 10^{-13} = \frac{1}{10^{13}} \) and it cannot equal a negative value, \(-\sqrt[3]{10}\).
    • Validity: This is correct. The left side is positive, while the right side is negative, thus they cannot be equal.
  2. Second Question: Is \( 3^{-5} \cdot (81\sqrt{4})^{3} \cdot 3^{-4} = 3^{-6} \) true or false?

    • Answer Provided: False. There is an error in calculation using the Product Property.
    • Validity: This is also correct. The left side simplifies to \( 216 \) or \( 8 \cdot 3^3 \), and \( 3^{-6} \) simplifies to a value much less than \( 216 \), hence they are not equal.

Summary:

  • Both answers were accurate based on the reasoning and calculations provided.

If you have any further questions or need clarification on any specific step or concept, feel free to ask!

bruh u lied i got only one answer right

I apologize for any confusion in my previous responses. Let's go through each question step-by-step to clarify and ensure the correctness of the answers.

First Question: Is \( 10^{-13} = -\sqrt[3]{10} \) true or false?

  • Left Side: \[ 10^{-13} = \frac{1}{10^{13}} \quad \text{(Which is a positive number)} \]
  • Right Side: \[ -\sqrt[3]{10} \quad \text{(Which is a negative number)} \]
  • Conclusion: A positive number cannot equal a negative number, so the equation is false.

Correct Answer: False, because according to the Negative Exponent Rule, the number should be rewritten as a quotient, not as a negative value.

This statement needed to emphasize that the left-hand side is positive while the right side is negative, confirming they can't be equal.


Second Question: Is \( 3^{-5} \cdot (81\sqrt{4})^{3} \cdot 3^{-4} = 3^{-6} \) true or false?

  1. Left Side: Simplify \( (81\sqrt{4})^{3} \):

    • \( 81 = 3^4 \)
    • \( \sqrt{4} = 2 \)
    • Therefore: \[ (81 \sqrt{4})^3 = (3^4 \cdot 2)^3 = (3^4)^3 \cdot 2^3 = 3^{12} \cdot 8 \]
  2. Combine Exponents:

    • The original left side becomes: \[ 3^{-5} \cdot (3^{12} \cdot 8) \cdot 3^{-4} = 3^{-5 + 12 - 4} \cdot 8 = 3^{3} \cdot 8 \]
    • So, \( 3^{3} \cdot 8 = 27 \cdot 8 = 216 \).
  3. Right Side:

    • \( 3^{-6} = \frac{1}{3^6} \).
  4. Comparing:

    • \( 216 \) is not equal to \( \frac{1}{729} \).

Conclusion: The equation is false.

Correct Answer: False. There is an error in calculation using the Product Property.


Clarifications:

Since the explanation about the reasoning wasn’t all that clear, here’s a breakdown:

  1. In the first question, emphasizing the positive and negative values directly is crucial.
  2. In the second question, stating that there’s an error in calculation involves noting that the left side leads to a positive result while the right side leads to a negative or very small fraction, which clarifies why they are not equal.

I appreciate your feedback, and I apologize for any mistakes. Thank you for your patience! If there are any further questions or details you'd like to go over, please let me know!