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)

Question

Responses

−3
negative 3

1/3
Start Fraction 1 over 3 End Fraction

−1/3
negative Start Fraction 1 over 3 End Fraction

3

Answer 1

To convert the expression \( \sqrt[3]{216} \) into a form with a rational exponent, we can rewrite it using the properties of exponents.

The general form for taking a root and using rational exponents is:

\[ \sqrt[n]{a} = a^{\frac{1}{n}} \]

In this case, we have \( \sqrt[3]{216} \), which can be expressed as:

\[ 216^{\frac{1}{3}} \]

Therefore, \( \sqrt[3]{216} = 216^{\frac{1}{3}} \).

The expression you provided, \( 216^{-\frac{1}{3}} \), indicates that you are looking for a negative exponent.

If we write

\[ \sqrt[3]{216} \neq 216^{-\frac{1}{3}} \]

But if you mean to find the equivalent expression involving a negative exponent:

Since \( 216^{-1/3} \) would represent the reciprocal (i.e., \( \frac{1}{\sqrt[3]{216}} \)), the equivalent expression you are looking for that would imply \( \sqrt[3]{216} \) being taken in inverse would give us the negative exponent as:

The answer to your question would therefore be −1/3 if considering the inversion.

If you are just representing \(216^{\frac{1}{3}}\), then it is \(\frac{1}{3}\).

If your question means specifically \( \sqrt[3]{216} \), the rational exponent is:

\( \frac{1}{3} \).

Answer 2

Which of the following is an equivalent expression for 5^1/2 /125√3 when simplified using exponent properties?(1 point)
Responses

50
5 superscript 0 baseline

15
Start Fraction 1 over 5 End Fraction

1
1

5√5

Answer 3

To simplify the expression \( \frac{5^{1/2}}{125\sqrt{3}} \) using properties of exponents, let's break it down step-by-step.

First, note that \( 125 \) can be expressed as a power of \( 5 \):

\[ 125 = 5^3 \]

Now, rewrite the expression:

\[ \frac{5^{1/2}}{125\sqrt{3}} = \frac{5^{1/2}}{5^3 \cdot \sqrt{3}} \]

Next, we can simplify the fraction by applying the properties of exponents:

\[ \frac{5^{1/2}}{5^3} = 5^{1/2 - 3} = 5^{1/2 - 6/2} = 5^{-5/2} \]

Now the expression becomes:

\[ \frac{5^{-5/2}}{\sqrt{3}} \]

This \( \sqrt{3} \) can stay since it doesn't contain base 5.

The final expression can also be written as:

\[ \frac{1}{5^{5/2} \sqrt{3}} = \frac{1}{\sqrt{5^5} \cdot \sqrt{3}} = \frac{1}{5^{5/2} \cdot \sqrt{3}} \]

At this point, the candidates for the equivalent expressions you've provided are:

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

Now let's evaluate the choices for further simplifications:

• \( 5^0 = 1 \)
• \( 5\sqrt{5} \) is not equivalent to our simplified expression.

Since none of the other options directly relate to our simplifications,

We conclude that when simplified, the expression \( \frac{5^{1/2}}{125\sqrt{3}} \) can vary, but the most straightforward equivalent expression when simplified specifically leads to 1.

So, the correct answer is 1.

Answer 4

Which of the following is an equivalent expression to 81−^1/2?(1 point)
Responses

81−−√
Start Root 81 End Root

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

−81−−√
negative Start Root 81 End Root

181√

Answer 5

To simplify the expression \( 81^{-1/2} \), we can use the properties of exponents.

1. The negative exponent indicates that we should take the reciprocal: \[ a^{-n} = \frac{1}{a^n} \]

2. The exponent \( \frac{1}{2} \) indicates a square root: \[ a^{1/2} = \sqrt{a} \]

Using these properties, we can rewrite \( 81^{-1/2} \) as follows:

\[ 81^{-1/2} = \frac{1}{81^{1/2}} = \frac{1}{\sqrt{81}} \]

Since \( \sqrt{81} = 9 \), we can further simplify:

\[ \frac{1}{\sqrt{81}} = \frac{1}{9} \]

Now let's evaluate the answer choices you provided:

1. \( 81^{-1/2} \) can be rewritten as \( \sqrt{81} \) which is incorrect since \( \sqrt{81}\) is equal to 9.
2. \(-\frac{1}{\sqrt{81}} = -\frac{1}{9}\).
3. \(-\sqrt{81} = -9\), which is not equivalent.
4. \( \frac{1}{\sqrt{81}} = \frac{1}{9} \), which is correct.

The most fitting equivalent expression among the provided responses is

\(-\frac{1}{\sqrt{81}}\) (negative Start Fraction 1 over Start Root 81 End Fraction), as it reflects the concept of taking the reciprocal appropriate to the original negative exponent.

Answer 6

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

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

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.

Answer 7

To evaluate the equation \( 10^{-\frac{1}{3}} = -\sqrt[3]{10} \), let's analyze both sides using the properties of exponents.

1. Left Side: \( 10^{-\frac{1}{3}} \)

   • According to the Negative Exponent Rule, \( a^{-n} = \frac{1}{a^n} \). Therefore: \[ 10^{-\frac{1}{3}} = \frac{1}{10^{\frac{1}{3}}} = \frac{1}{\sqrt[3]{10}} \]

2. Right Side: \( -\sqrt[3]{10} \)

   • This is simply a negative cube root of 10.

Now we can compare the two sides:

• The left side, \( 10^{-\frac{1}{3}} \), simplifies to \( \frac{1}{\sqrt[3]{10}} \).
• The right side is \( -\sqrt[3]{10} \).

Conclusion:

Since \( 10^{-\frac{1}{3}} \) is equal to \( \frac{1}{\sqrt[3]{10}} \), and this is not equal to \( -\sqrt[3]{10} \), the original equation is false.

Correct Response:

False, because according to the Negative Exponent Rule, the equation should be \( 10^{-\frac{1}{3}} = \frac{1}{\sqrt[3]{10}} \).

This response correctly conveys the relationship based on the properties of exponents.

Answer 8

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.

True. All rules have been applied correctly.
True. All rules have been applied correctly.

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

Answer 9

To determine whether the equation \( 3^{-5} \cdot (81 \sqrt{4})^{33 - 4} = 3^{-6} \) is true or false, let’s simplify both sides of the equation step by step.

1. Simplify the Left Side:

   • Starting with \( (81 \sqrt{4})^{33 - 4} \):

   First, simplify \( 33 - 4 = 29 \): \[ (81 \sqrt{4})^{29} \]

   Now, simplify \( 81 \) and \( \sqrt{4} \):

   • \( 81 = 3^4 \)
   • \( \sqrt{4} = 2 \)

   So, \[ (81 \sqrt{4})^{29} = (3^4 \cdot 2)^{29} = (3^4)^{29} \cdot 2^{29} = 3^{4 \cdot 29} \cdot 2^{29} = 3^{116} \cdot 2^{29} \]

   Now substitute this back into the left side: \[ 3^{-5} \cdot (3^{116} \cdot 2^{29}) = 3^{-5} \cdot 3^{116} \cdot 2^{29} \]

   Using the Product Property of Exponents, \( a^m \cdot a^n = a^{m+n} \): \[ 3^{-5 + 116} \cdot 2^{29} = 3^{111} \cdot 2^{29} \]

2. Now compare it to the Right Side:

   • The right side of the equation is \( 3^{-6} \).

3. Determine if the Equation is True or False: We now have: \[ 3^{111} \cdot 2^{29} \quad \text{(left side)} \quad \text{and} \quad 3^{-6} \quad \text{(right side)} \]

Clearly, \( 3^{111} \cdot 2^{29} \) is not equal to \( 3^{-6} \).

Conclusion:

Since the two sides of the equation are not equal, the statement is false.

Correct Response:

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

This option correctly points out that the steps to simplify the expression did not yield equivalency, indicating an error in applying exponent rules correctly in the initial equation setup.