To simplify the expression [tex]\(\sqrt{50} + \sqrt{2}\)[/tex], let's follow a step-by-step process.
1. Simplify [tex]\(\sqrt{50}\)[/tex]:
- We start by breaking down [tex]\(50\)[/tex] into its prime factors: [tex]\(50 = 25 \times 2\)[/tex].
- Therefore, [tex]\(\sqrt{50} = \sqrt{25 \times 2}\)[/tex].
- Recognizing that [tex]\(\sqrt{25}\)[/tex] is a perfect square, we have [tex]\(\sqrt{25 \times 2} = \sqrt{25} \times \sqrt{2} = 5 \times \sqrt{2}\)[/tex].
Consequently, [tex]\(\sqrt{50}\)[/tex] simplifies to [tex]\(5\sqrt{2}\)[/tex].
2. Simplify [tex]\(\sqrt{2}\)[/tex]:
- [tex]\(\sqrt{2}\)[/tex] is already in its simplest form as it cannot be simplified further.
3. Combine the simplified terms:
- Now we combine the terms we obtained: [tex]\(5\sqrt{2}\)[/tex] and [tex]\(\sqrt{2}\)[/tex].
- Since both terms include [tex]\(\sqrt{2}\)[/tex], we can add them directly: [tex]\(5\sqrt{2} + \sqrt{2} = (5 + 1)\sqrt{2} = 6\sqrt{2}\)[/tex].
So, the simplified form of the original expression [tex]\(\sqrt{50} + \sqrt{2}\)[/tex] is [tex]\(6\sqrt{2}\)[/tex].
However, if we were to express the numerical or decimal value of the sum:
- [tex]\(\sqrt{50} \approx 7.0710678118654755\)[/tex]
- [tex]\(\sqrt{2} \approx 1.4142135623730951\)[/tex]
- Summing these, we get [tex]\(7.0710678118654755 + 1.4142135623730951 \approx 8.485281374238571\)[/tex]
So the decimal approximation of [tex]\(\sqrt{50} + \sqrt{2}\)[/tex] is approximately [tex]\(8.485281374238571\)[/tex].
