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1. Using the properties of exponents to generate an equivalent expression, which of the following would be the rational exponent of the equivalent expression to 216−−−√3

if it were written in exponential form instead of radical form?(1 point)
Responses

−13
negative 1 third

13
1 third

-3
-3

3
3
Question 2
2. Which of the following is an equivalent expression for 512125√3
when simplified using exponent properties?(1 point)
Responses

5√5
the fraction with numerator square root of 5 and denominator 5

1
1

50
5 to the 0 power

15
1 fifth
Question 3
3. Which of the following is an equivalent expression to 81−12
?(1 point)
Responses

−181√
negative the fraction with numerator 1 and denominator square root of 81

−81−−√
negative square root of 81

81−−√
square root of 81

181√
the fraction with numerator 1 and denominator square root of 81
Question 4
4. 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

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 raised to the negative 1 third power is equal to negative the fraction with numerator 1 and denominator cube root of 10

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.
Question 5
5. 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. The numerator simplifies to 3−2
and 3−23−4≠3−6
True. The numerator simplifies to 3 to the negative 2 power and the fraction with numerator 3 to the negative 2 power and denominator 3 to the negative 4 power is not equal to 3 to the negative 6 power

False. The numerator simplifies to 3−2
and 3−23−4≠3−6
False. The numerator simplifies to 3 to the negative 2 power and the fraction with numerator 3 to the negative 2 power and denominator 3 to the negative 4 power is not equal to 3 to the negative 6 power

True. (81−−√4)3=32
and 3−103−4=3−6
True. open paren 4th root of 81 close paren cubed is equal to 3 squared and the fraction with numerator 3 to the negative 10 power and denominator 3 to the negative 4 power is equal to 3 to the negative 6 power

False. (81−−√4)3
cannot be determined
False. open paren 4th root of 81 close paren cubed cannot be determined
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1 answer

Let's go through each question one by one.

Question 1:

Generate an equivalent expression for \( \sqrt[3]{216} \) in exponential form.

To express \( \sqrt[3]{216} \) in exponential form, we recognize it as \( 216^{1/3} \). Since \( 216 = 6^3 \), we can rewrite it as \( (6^3)^{1/3} = 6^{3 \cdot (1/3)} = 6^1 = 6 \). Therefore, the rational exponent is:

  • \( \frac{1}{3} \)

However, for \( 216^{-1/3} \), which is what the equation states:

  • The exponent is \( -\frac{1}{3} \).

Response:

  • −1/3

Question 2:

Simplify \( 512125\sqrt{3} \).

To simplify the expression \( 512125\sqrt{3} \):

  1. You need to express \( 512125 \) in exponential form. If we break it down:

    • \( 512 = 2^9 \)
    • So, \( \sqrt{3} = 3^{1/2} \)

    Therefore, combining gives us: \[ 512 \cdot 125 \cdot 3^{1/2} = 2^9 \cdot 5^3 \cdot 3^{1/2} \]

In improper fractional simplification, this can lead to 5’s division.

Response:

  • Since simplifications may vary, the closest logical answer is:
  • 1

Question 3:

Equivalent expression to \( 81^{-1/2} \)

The expression \( 81^{-1/2} \):

  • Can be rewritten using the negative exponent rule:
  • \( 81^{-1/2} = \frac{1}{81^{1/2}} = \frac{1}{\sqrt{81}} = \frac{1}{9} \).

Response:

  • 1/81^{1/2}.

Question 4:

True or false: \( 10^{-1/3} = -10\sqrt[3]{3} \)

First, \( 10^{-1/3} \) is \( \frac{1}{10^{1/3}} \) or \( \frac{1}{\sqrt[3]{10}} \). The right-hand side \( -10\sqrt[3]{3} \) has a negative sign and does not match these forms exactly, therefore:

Response:

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

Question 5:

Is the equation \( 3^{-5} \cdot (81^{\frac{1}{4}})^{3} \cdot 3^{-4} = 3^{-6} \) true or false?

To simplify \( 3^{-5} \cdot (81^{1/4})^{3} \cdot 3^{-4} \):

  1. \( (81^{1/4})^{3} = 81^{3/4} = (3^4)^{3/4} = 3^{3} \).
  2. Now combine:
  • \( 3^{-5} \cdot 3^{3} \cdot 3^{-4} = 3^{-5 + 3 - 4} = 3^{-6} \).

Thus, we find:

  • \( 3^{-5} \cdot (81^{1/4})^{3} \cdot 3^{-4} = 3^{-6} \).

Response:

  • True. The numerator simplifies to \( 3^{-2} \) and \( 3^{-2} \cdot 3^{-4} = 3^{-6} \).

Feel free to ask for clarifications or more details for any item!