Which of the following simplifies to a negative number? (1 point) Responses Image with alt text: negative 4 to the negative 4 Image with alt text: the quantity negative 4 to the negative 4 Image with alt text: the 4 to the negative 4
The expression "the quantity negative 4 to the negative 4" simplifies to a negative number.
To determine which of the following simplifies to a negative number, we need to evaluate each expression.
1. Image with alt text: negative 4 to the negative 4
To simplify this expression, we need to apply the exponent rule, which states that a negative number raised to an even power will result in a positive number. Since -4 is raised to the power of -4, which is even, the result will be positive. So, it does not simplify to a negative number.
2. Image with alt text: the quantity negative 4 to the negative 4
This expression involves raising a negative number, -4, to a negative power, -4. When we have a negative base raised to a negative exponent, we need to use the exponent rule, which states that dividing 1 by the positive base raised to the absolute value of the negative exponent will result in a positive number. In this case, (-4)^(-4) simplifies to 1/((-4)^4), which is positive. So, it does not simplify to a negative number.
3. Image with alt text: the 4 to the negative 4
This expression involves raising a positive base, 4, to a negative exponent, -4. According to the exponent rule, when we have a positive base raised to a negative exponent, we can rewrite the expression as 1/(base raised to the positive exponent). So, 4^(-4) simplifies to 1/(4^4), which is positive. Therefore, it does not simplify to a negative number.
In conclusion, none of the expressions simplify to a negative number.
To determine which of the given expressions simplifies to a negative number, we need to evaluate each expression.
1. Image with alt text: negative 4 to the negative 4
To simplify negative 4 to the negative 4, we need to calculate (-4)^(-4).
Using the rules of exponents, a negative number raised to an even power will always result in a positive number.
So, (-4)^(-4) simplifies to a positive number.
2. Image with alt text: the quantity negative 4 to the negative 4
In this expression, the negative exponent is applied to the entire quantity of -4.
To determine the value, we need to calculate (-4)^(-4).
Again, using the rules of exponents, the negative exponent will result in the reciprocal of the base raised to the positive exponent. Therefore, (-4)^(-4) simplifies to a positive number.
3. Image with alt text: the 4 to the negative 4
To simplify 4 to the negative 4, we calculate 4^(-4).
Applying the same rule of exponents, a positive number raised to a negative exponent will result in a fraction with 1 as the numerator and the base raised to the positive exponent as the denominator.
Therefore, 4^(-4) simplifies to a positive fraction (1/4^4) or 1/256.
Therefore, none of the given expressions simplifies to a negative number.