Question

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

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. - not selected, this is the correct answer

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. - no response given

True, because the negative sign moves to the front when writing an equivalent expression.
Show me how to solve

Answer 1

To determine whether the equation \( 10^{-1/3} = -\sqrt[3]{10} \) is true or false, we can simplify and analyze both sides of the equation.

**Step 1: Simplify the left side.**

The left side is \( 10^{-1/3} \). According to the Negative Exponent Rule:

\[
a^{-b} = \frac{1}{a^b}
\]

So, we can rewrite \( 10^{-1/3} \) as:

\[
10^{-1/3} = \frac{1}{10^{1/3}} = \frac{1}{\sqrt[3]{10}}
\]

**Step 2: Evaluate the right side.**

The right side of the equation is \( -\sqrt[3]{10} \).

**Step 3: Compare the two sides.**

Now we have:

\[
10^{-1/3} = \frac{1}{\sqrt[3]{10}} \quad \text{(left side)}
\]
\[
-\sqrt[3]{10} \quad \text{(right side)}
\]

**Step 4: Analyze the values.**

We can see:
- The left side \( \frac{1}{\sqrt[3]{10}} \) is a positive number because \(\sqrt[3]{10}\) is a positive number (any real cube root is positive).
- The right side \( -\sqrt[3]{10} \) is a negative number, as it has a negative sign in front.

Since the left side is positive and the right side is negative, the two sides cannot be equal.

**Conclusion:**

The equation \( 10^{-1/3} = -\sqrt[3]{10} \) is **false**. This is because the Negative Exponent Rule allows us to rewrite \( 10^{-1/3} \) as \( \frac{1}{\sqrt[3]{10}} \), which is positive and cannot equal the negative value \( -\sqrt[3]{10} \).

**Selected response:**

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