Question

Express using positive and negative exponents: f(x) = (-2x^3 + 3x + 9)^(1/2).

Answer 1

To express the function f(x) = (-2x^3 + 3x + 9)^(1/2) using positive and negative exponents, apply the rules of exponents and distribute the exponent. Simplify each term and combine them to get the expression f(x) = 2i * x^(3/2) + 3x + 9.

Step-by-step explanation:

To express the function f(x) = (-2x^3 + 3x + 9)^(1/2) using positive and negative exponents, we need to understand the rules of exponents. In this case, the exponent of 1/2 represents the square root. To simplify:

1. Start by distributing the exponent to each term inside the parentheses:
   -2x^3 raised to the 1/2 power is equal to (-2x^3)^(1/2).
2. Apply the rule that for any number a, (a^n)^m = a^(n*m):
   (-2x^3)^(1/2) is equal to (-2)^(1/2) * (x^3)^(1/2).
3. Use the rule that the square root of a negative number is equal to the square root of its absolute value multiplied by i:
   (-2)^(1/2) equals √((-2)^2) * i = √4 * i = 2i.
4. Simplify the expression for x^3 raised to the 1/2 power:
   (x^3)^(1/2) is equal to √(x^3) = x^(3/2).

Putting it all together, the expression can be written as:
f(x) = 2i * x^(3/2) + 3x + 9.