Radicals & Rational Exponents Quick Check

Question

4 of 54 of 5 Items
Question
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)
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 .

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.
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.
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Answer 1

To determine whether the equation \( 10^{-13} = -10^{-\sqrt{3}} \) is true or false, we can analyze both sides.

1. Left Side: \[ 10^{-13} = \frac{1}{10^{13}} \] This is a standard application of the negative exponent rule.

2. Right Side: \[ -10^{-\sqrt{3}} = -\frac{1}{10^{\sqrt{3}}} \] Again, applying the negative exponent rule, we rewrite \( 10^{-\sqrt{3}} \).

Now we will compare both sides:

• The left side \( \frac{1}{10^{13}} \) represents a positive value because it is the reciprocal of a positive number.
• The right side \( -\frac{1}{10^{\sqrt{3}}} \) is negative because it is the negative of a positive value.

Since one side is positive and the other side is negative, we can conclude that:

False, because the left side is positive while the right side is negative, making them unequal.

Among the provided options, the closest correct response would be:

False, because according to the Negative Exponent Rule, the equation should be \( 10^{-13} = -\frac{1}{10^{\sqrt{3}}} \).

Please choose the response that aligns most closely with this explanation.