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.

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 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.