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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

-3
-3

−13
negative 1 third

3
3

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

1
1

50
5 to the 0 power

15
1 fifth

5√5
the fraction with numerator square root of 5 and denominator 5
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−−√
square root of 81

−81−−√
negative 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

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

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
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. (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

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

1 answer

Let's go through each question one by one:

Question 1:

  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 \(216^{-\frac{1}{3}}\) if it were written in exponential form instead of radical form?
  • The expression \(216^{-\frac{1}{3}}\) means the same as \(-\sqrt[3]{216}\), but in exponential form, it is already given as \(216^{-\frac{1}{3}}\). Thus the rational exponent is indeed \(-\frac{1}{3}\).

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

Question 2:

  1. Which of the following is an equivalent expression for \(512^{\frac{1}{25}}\) when simplified using exponent properties?
  • We can express \(512\) as \(2^9\) (because \(2^9 = 512\)). Then, we have \((2^9)^{\frac{1}{25}} = 2^{\frac{9}{25}}\). This does not match the response options directly, but looking at the options again will clarify that we have \((2^{\frac{9}{25}}) \neq 1, 5^0, 15, 5\sqrt{5}\). Since all options do not seem to apply directly, you would need to select the option that best aligns with the available responses—none directly fit, but \(5^0 = 1\) is just always true in exponent terms.

Answer: 1

Question 3:

  1. Which of the following is an equivalent expression to \(81^{- \frac{1}{2}}\)?
  • The expression \(81^{- \frac{1}{2}}\) can be rewritten as \(\frac{1}{\sqrt{81}} = \frac{1}{9}\).

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

Question 4:

  1. Use the properties of exponents to determine whether the equation \(10^{- \frac{1}{3}} = -10^{\frac{1}{3}}\) is true or false.
  • The left side, \(10^{- \frac{1}{3}}\), is the same as \(\frac{1}{10^{\frac{1}{3}}}\), while the right side is the negative of the cube root of \(10\). They are not equivalent.

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

Question 5:

  1. Is the equation \(3^{-5} \cdot (81^{\frac{1}{4}})^{3} \cdot 3^{-4} = 3^{-6}\) true or false?
  • Let's simplify the left side:
    1. \((81^{\frac{1}{4}})^{3} = 81^{\frac{3}{4}} = (3^4)^{\frac{3}{4}} = 3^3\)
    2. Therefore the left side becomes \(3^{-5} \cdot 3^3 \cdot 3^{-4}\)
    3. By adding the exponents: \(-5 + 3 - 4 = -6\)
  • Thus the left side simplifies to \(3^{-6}\), which equals the right side.

Answer: True. \((81^{\frac{1}{4}})^3 = 3^2\) and \(3^{-5} \times 3^3 \times 3^{-4} = 3^{-6}\)

Let me know if you need any further assistance!