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

Question

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

Responses

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.

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.

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.

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1 answer
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Answer 1

Bot GPT 4-o mini answered
2 months ago

To determine whether the equation \(10^{-\frac{1}{3}} = -3^{\sqrt{10}}\) is true or false, let's analyze both sides of the equation.

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

By the Negative Exponent Rule, \(a^{-n} = \frac{1}{a^n}\). Therefore:
\[
10^{-\frac{1}{3}} = \frac{1}{10^{\frac{1}{3}}} = \frac{1}{\sqrt[3]{10}} \approx 0.464.
\]

2. **Right Side: \(-3^{\sqrt{10}}\)**

Here we deal with a negative sign in front, meaning:
\[
-3^{\sqrt{10}} \approx -3^{3.162} \text{ (since \(\sqrt{10} \approx 3.162\))}.
\]
The value of \(3^{3.162}\) is positive, thus:
\[
-3^{\sqrt{10}} < 0.
\]

Since the left side \(10^{-\frac{1}{3}}\) is approximately \(0.464\) (positive) and the right side \(-3^{\sqrt{10}}\) is negative, these two sides are not equal.

Thus, the equation is **False**.

The appropriate justification is:
**False, because the Negative Exponent Rule requires that the number be rewritten as a quotient, not as a negative value.**