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 according to the Negative Exponent Rule, the equation should be 10^−1/3=−1/3√.10
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 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 according to the Negative Exponent Rule, the equation should be 10^−1/3=−1/3√10.
Justification: According to the Negative Exponent Rule, a negative exponent can be rewritten as the reciprocal of the base raised to the positive exponent. In this case, the base is 10 and the exponent is -1/3. So, the equation should be 10^(-1/3) = 1/(√10)^3 = 1/(∛10) = -1/3√10. Therefore, the given equation is false.
If 3√10 means ∛10, then the equation is false (even after using appropriate parentheses), since
10^(-1/3) = 1/10^(1/3) = 1/∛10
You are correct. I apologize for the mistake in my previous response. The correct justification is:
False, because according to the Negative Exponent Rule, the equation should be 10^(-1/3) = 1/∛10.
Justification: According to the Negative Exponent Rule, a negative exponent means the reciprocal of the base raised to the positive exponent. In this case, the base is 10 and the exponent is -1/3. So, the equation should be 10^(-1/3) = 1/(∛10). Therefore, the given equation is false.
False, because according to the Negative Exponent Rule, the equation should be 10^−1/3=−1/3√.10
False, because according to the Negative Exponent Rule, the equation should be 10^-1/3 = -1/3√10.
To determine whether the equation 10^(-1/3) = -3√10 is true or false, we can use the properties of exponents. The Negative Exponent Rule states that when a number is raised to a negative exponent, it can be rewritten as the reciprocal of the number raised to the positive exponent. In this case, 10^(-1/3) can be rewritten as 1/(10^(1/3)).
To simplify this expression further, we need to evaluate the cube root of 10^(1/3). The cube root of a number can be written as the number raised to the exponent of 1/3.
So, the cube root of 10^(1/3) is (10^(1/3))^(1/3), which simplifies to 10^(1/9).
Therefore, the equation becomes 1/(10^(1/3)) = -3√10.
Comparing this with the given equation, we can see that the expression on the left-hand side is not equal to the expression on the right-hand side. Hence, the equation 10^(-1/3) = -3√10 is false.
The appropriate justification is false, because according to the Negative Exponent Rule, the equation should be 10^(-1/3) = -1/3√10.