What is /32 expressed in simplest radical form?
The Correct Answer and Explanation is :
To express (\sqrt{32}) in its simplest radical form, we first need to simplify the square root. Let’s go through the process step-by-step:
- Factor 32 into prime factors:
[
32 = 2^5
] - Find the square root of 32:
The square root of a product can be written as the product of the square roots. In other words:
[
\sqrt{32} = \sqrt{2^5}
] - Simplify the square root:
We can break down the square root as follows:
[
\sqrt{2^5} = \sqrt{2^4 \times 2} = \sqrt{2^4} \times \sqrt{2} = 4\sqrt{2}
]
Here, (\sqrt{2^4} = 4) because (2^4 = 16), and (\sqrt{16} = 4), while (\sqrt{2}) remains as is.
Thus, the simplest radical form of (\sqrt{32}) is:
[
\boxed{4\sqrt{2}}
]
Explanation:
To simplify a square root like (\sqrt{32}), the goal is to find a perfect square factor of 32 and take the square root of that factor. In this case, (2^4) is a perfect square, and when we take its square root, it simplifies to 4. We then multiply this by the square root of the leftover factor, which is (\sqrt{2}), to get the final answer of (4\sqrt{2}).
This process of simplifying square roots helps make expressions involving square roots more manageable and easier to work with in mathematical operations like addition, subtraction, or multiplication of square roots. Additionally, it ensures that the expression is in its simplest form, which is generally expected in algebraic contexts.
I will now generate an image to visually represent this simplification process.
Here is a visual representation of the step-by-step process to simplify (\sqrt{32}) to (4\sqrt{2}). The image shows the breakdown of 32 into its prime factors, the simplification of the square root, and the final expression.
Let me know if you need further clarification!