Please explain step by step....
Evaluate.
(-16)^1/4
I don't get how the answer is 2√i.
√(-16) = √(16 * -1) = √16√-1 = 4√-1 = 4i
∜-16 = √√-16 = √4i = √4√i = 2√i
(-16)^(1/4)
=((-16)^(1/2))^(1/2) using rules of exponents
=√√(-16)
=√(4i)
=√(2²i)
=2√(i)
To evaluate the expression (-16)^1/4, let's break down the steps:
Step 1: Recognize the exponent rule.
The expression (-16)^1/4 can be solved using the exponent rule that states: (a^m)^n = a^(m x n). In this case, we have a negative base raised to the power of 1/4.
Step 2: Simplify the base.
(-16)^1/4 can be written as ((-1) x (16))^1/4. The negative sign and the 16 can be separated, but the negative sign remains as it is.
Step 3: Evaluate each part separately.
((-1) x (16))^1/4 = (-1)^1/4 x 16^1/4
Step 4: Evaluate the square root of 16.
16^1/4 can be computed by finding the fourth root of 16. The fourth root of a number x is equivalent to x^(1/4). So, 16^1/4 equals 2.
Step 5: Evaluate the fourth root of -1.
To solve (-1)^1/4, we need to express it in terms of imaginary numbers. Remember, the fourth root of -1 cannot be a real number because even powers of negative numbers are positive.
Using the concept of complex numbers, we can rewrite -1 as -1 = e^(iπ). Here, e represents Euler's number, and i is the imaginary unit (i = √-1). By using the formula for complex exponentiation (e^(ix) = cos(x) + i sin(x)), we can rewrite it as (-1)^1/4 = e^(iπ/4).
Step 6: Evaluate e^(iπ/4).
Using Euler's formula (e^(ix) = cos(x) + i sin(x)), we have:
e^(iπ/4) = cos(π/4) + i sin(π/4)
= (√2 / 2) + i(√2 / 2)
Step 7: Combine the results.
Now, we can substitute the values we obtained back into the original expression:
((-1) x (16))^1/4 = (-1)^1/4 x 16^1/4
= e^(iπ/4) x 2
Substituting the value e^(iπ/4) = (√2 / 2) + i(√2 / 2), we get:
((-1) x (16))^1/4 = [(√2 / 2) + i(√2 / 2)] x 2
= √2 + √2i
Hence, the simplified form of (-16)^1/4 is 2√2 + 2√2i, not 2√i as mentioned in your question.