True or false math problem?
Is the following statement always, sometimes, or never true? Explain your choice.
A number raised to a negative exponent is negative.
Update:Understood: For example, 9^-7 would be 1/9^7, which would be 1/4782969, which is positive.
Comments
NO!
Take a look at these examples.
1. 2^-2 = 1/2^2 = 1/4
2. 4^-4 = 1/4^4 = 1/256
A number raised to a negative exponent is negative.
False
Because, a number raised to a negative exponent is always a fraction:
Example 3^-1 = 1/3
(4/5)^-1 = 5/4
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Best of luck.
If the number is a positive number, such as, 4, then 4^(-1) = 0.25
If the number is a negative number such as, -4, then (-4)^(-1) = -025
So the statement is "sometimes" true.
If the base is positive and the exponent is negative then the answer will be positive if the the base is negative and the exponent it negative then the answer will be negative so sometimes
Case 1: (pos N)^(neg M) = positive # ... where N and M are any real numbers
Example: (+3.4)^(-2.7) = +0.0367 ... positive result ◀◀
~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~
Case 2: (neg N)^(neg M) = 1 ⁄ [ (neg N)^(pos M) ] ... where N and M are integers
if M is EVEN, then: (neg N)^(neg even M) = positive result ◀◀
Example: (-3)^(-4) = +0.0123
if M is ODD, then: (neg N)^(neg odd M) = negative result ◀◀
Example: (-3)^(-3) = -0.03781
For fractional M values, the result is a complex (imaginary) value.
Example: (-3)^(-2.3) = 0.047 – 0.06465 i ... complex number result ◀◀
~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~
Also, FYI ...
(posN)^(0) = 1
(neg N)^(0) = 1
Answer: SOMETIMES ◀◀