Simplify f(0) square root of 4(0)+12

Solution:

Step 1:

Apply the function $f$ to $0$ and distribute the multiplication: $0\cdot \sqrt{4(0)+12}$.

Step 2:

Calculate the product of $4$ and $0$: $0\cdot \sqrt{0+12}$.

Step 3:

Combine the terms inside the square root: $0\cdot \sqrt{12}$.

Step 4:

Express $12$ as a product of its prime factors.

Step 4.1:

Extract the square factor from $12$: $0\cdot \sqrt{4\cdot 3}$.

Step 4.2:

Represent $4$ as $2^2$: $0\cdot \sqrt{2^2\cdot 3}$.

Step 5:

Simplify the square root by taking out the square term: $0\cdot (2\cdot \sqrt{3})$.

Step 6:

Final multiplication of $0$ with the simplified radical.

Step 6.1:

Multiply $2$ by $0$: $0\cdot \sqrt{3}$.

Step 6.2:

Multiply $0$ by $\sqrt{3}$: $0$.

Knowledge Notes:

The problem given is to simplify the expression $f(0)\cdot \sqrt{4(0)+12}$. This involves several steps of algebraic manipulation, specifically dealing with square roots and multiplication by zero.

1. Multiplication by Zero: Any number multiplied by zero is zero. This is a fundamental property of multiplication in arithmetic.

2. Square Roots: The square root of a number is a value that, when multiplied by itself, gives the original number. The square root of a perfect square (like $4$ which is $2^2$) is an integer.

3. Prime Factorization: This is the process of breaking down a composite number into its prime factors. For example, $12$ can be factored into $2^2\times 3$.

4. Simplifying Square Roots: When a number under a square root can be expressed as a product of a square number and another number, the square root of the square number can be taken out of the radical, simplifying the expression.

5. Algebraic Simplification: This involves combining like terms and simplifying expressions according to the rules of algebra.

In this problem, since the function $f$ is multiplied by zero, the entire expression simplifies to zero, regardless of the other terms. This is because the multiplication by zero overrides any other calculations within the expression.