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)

Question

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 .

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.

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

Answer 1

To determine the validity of the equation \( 10^{-13} = -10^{-\frac{1}{3}} \), we need to analyze both sides using the properties of exponents.

The left side is: \[ 10^{-13} = \frac{1}{10^{13}}. \]

The right side can be rewritten using the Negative Exponent Rule: \[ -10^{-\frac{1}{3}} = -\frac{1}{10^{\frac{1}{3}}}. \]

This means the equation becomes: \[ \frac{1}{10^{13}} = -\frac{1}{10^{\frac{1}{3}}}. \]

Clearly, \( \frac{1}{10^{13}} \) is a positive number, while \( -\frac{1}{10^{\frac{1}{3}}} \) is a negative number. Thus, these two sides are not equal.

Therefore, the correct response is:

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

Answer 2

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 cubedcannot be determined.

Answer 3

Let's analyze and simplify the left side of the equation \( 3^{-5} \cdot (81^{\frac{1}{4}})^3 \cdot 3^{-4} \) to see if it's equal to \( 3^{-6} \).

Step 1: Simplifying \( (81^{\frac{1}{4}})^3 \)

First, we calculate \( 81^{\frac{1}{4}} \): \[ 81 = 3^4 \implies 81^{\frac{1}{4}} = (3^4)^{\frac{1}{4}} = 3^{4 \cdot \frac{1}{4}} = 3^1 = 3. \] Now substituting back, we have: \[ (81^{\frac{1}{4}})^3 = 3^3 = 27. \] So, our left side becomes: \[ 3^{-5} \cdot 3^3 \cdot 3^{-4}. \]

Step 2: Combining the Exponents

Using the properties of exponents: \[ 3^{-5} \cdot 3^3 \cdot 3^{-4} = 3^{-5 + 3 - 4} = 3^{-6}. \]

Conclusion

Now we have simplified the left-hand side to \( 3^{-6} \), which is exactly equal to the right-hand side \( 3^{-6} \).

Result

The statement: \[ 3^{-5} \cdot (81^{\frac{1}{4}})^3 \cdot 3^{-4} = 3^{-6} \] is True.

Thus, the correct response is:

True. (81^{\frac{1}{4}})^3 = 3^2 and \( 3^{-10}\) / \(3^{-4} = 3^{-6} \).