Simplify the surd square root 12 x square root 8
The Correct Answer and Explanation is :
To simplify ( \sqrt{12} \times \sqrt{8} ), we can use the property that (\sqrt{a} \times \sqrt{b} = \sqrt{a \times b}). So,
[
\sqrt{12} \times \sqrt{8} = \sqrt{12 \times 8}
]
Step-by-Step Solution
- Multiply the Numbers Under the Surds:
[
12 \times 8 = 96
]
So, (\sqrt{12} \times \sqrt{8} = \sqrt{96}). - Simplify the Square Root of 96:
To simplify (\sqrt{96}), we need to break it down into a product of factors, ideally one of which is a perfect square. Let’s factorize 96:
[
96 = 16 \times 6
]
Since 16 is a perfect square, we can rewrite (\sqrt{96}) as:
[
\sqrt{96} = \sqrt{16 \times 6} = \sqrt{16} \times \sqrt{6} = 4\sqrt{6}
]
Final Answer
[
\sqrt{12} \times \sqrt{8} = 4\sqrt{6}
]
Explanation
In this simplification process, we used two main properties of square roots: the multiplication property (\sqrt{a} \times \sqrt{b} = \sqrt{a \times b}) and the ability to break down a square root into simpler components if we have a perfect square as a factor.
By rewriting ( \sqrt{12} \times \sqrt{8} ) as (\sqrt{96}) and then factoring 96 into (16 \times 6), we could take the square root of 16 directly (since (\sqrt{16} = 4)) and leave (\sqrt{6}) as it is because 6 isn’t a perfect square. This approach allows us to express the product in its simplest surd form: (4\sqrt{6}).