Simplify f(0) square root of 4(0)+12
The problem is asking to evaluate the mathematical function f at the argument 0, and then take the square root of the result of that evaluation multiplied by 4(0) plus 12. In simpler terms, you need to calculate what f(0) is, then multiply that by the value you get from 4 times 0, and add 12 to that product. Afterward, you'll need to find the square root of the sum you've obtained.
$f \left(\right. 0 \left.\right) \sqrt{4 \left(\right. 0 \left.\right) + 12}$
Answer
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.
Multiplication by Zero: Any number multiplied by zero is zero. This is a fundamental property of multiplication in arithmetic.
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.
Prime Factorization: This is the process of breaking down a composite number into its prime factors. For example, $12$ can be factored into $2^2 \times 3$.
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.
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.
