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

Question

?(1 point)
Responses

-3
-3

13
1 third

3
3

−13
negative 1 third
Question 2
Which of the following is an equivalent expression for 512125√3
when simplified using exponent properties?(1 point)
Responses

50
5 to the 0 power

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

1
1

15
1 fifth
Question 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

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

81−−√
square root of 81
Question 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 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

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.

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
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

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

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
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Answer 1

Let's go through each question one by one.

Question 1

Which of the following would be the rational exponent on an expression equivalent to \( 216^{-\frac{1}{3}} \)?

• The expression \( 216^{-\frac{1}{3}} \) can be rewritten using the properties of exponents. The negative exponent indicates that we take the reciprocal, while the \(\frac{1}{3}\) exponent signifies the cube root.
• Therefore, \( 216^{-\frac{1}{3}} = \frac{1}{216^{\frac{1}{3}}} \).

Thus, the answer is: −13 (negative one third)

Question 2

Which of the following is an equivalent expression for \( 512\sqrt[3]{125} \) when simplified using exponent properties?

• We can simplify \( 512 \) and \( \sqrt[3]{125} \).
• \( 512 = 2^9 \) and \( \sqrt[3]{125} = 5^{\frac{5}{3}} \).
• The combination gives \( 2^9 \times 5^{\frac{5}{3}} \), but that doesn’t match the options.
• However, since \( \sqrt[3]{125} = 5^{\frac{5}{3}} \) and \( 5^{\frac{5}{3}} = 5 \cdot 5^{\frac{2}{3}} = 5^{1} \sqrt[3]{25}\), and simplifying fully is intricate without a specific target.

However, given the choices seem to stray from pure expression reliance, and without clarity— a rough option arising from \(5^{5/2} \).

The most logical choice here remains unclear but likely option implying 5\sqrt{5} more closely resembles those estimations. However, straight simplification from the given format suggests 50 (5^0) is a clearer fallback.

Question 3

Which of the following is an equivalent expression to \( 81^{-1/2} \)?

• \( 81^{-1/2} = \frac{1}{81^{1/2}} = \frac{1}{\sqrt{81}} = \frac{1}{9} \).

Thus, the answer is: \( 181\sqrt{} \) (the fraction with numerator 1 and denominator square root of 81)

Question 4

Use the properties of exponents to determine whether the equation \( 10^{-1/3} = -10^{-\sqrt{3}} \) is true or false.

• The left side, \( 10^{-1/3} = \frac{1}{10^{1/3}} \), while the right side suggests \( - \frac{1}{10^{\sqrt{3}}} \), leading to dissimilar results.

Thus, the answer is: False, because according to the Negative Exponent Rule, the equation should be \( 10^{-1/3} = -\frac{1}{10^{1/3}} \).

Question 5

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

• The term \( (81^{\frac{1}{4}})^{3} = 81^{\frac{3}{4}} \) can be expressed as \( (3^4)^{\frac{3}{4}} = 3^{3} = 27 \).
• Replacing gives \( 3^{-5} \cdot 27 \cdot 3^{-4} \) which simplifies to collect powers of \( 3 \), yielding:

Upon figuring,

• Calculating it yields \( 3^{-5-4+3} = 3^{-6} \).

Thus, the answer is: True. The numerator simplifies to \( 3^{-6} \) (= \( \frac{1}{3^6} \)).

Feel free to ask further if you need clarification or check on these!