To simplify the expression [tex]\(8 \sqrt{10} + 3 \sqrt{360}\)[/tex], let's follow a step-by-step approach:
1. Identify and Simplify the Terms:
- The first term, [tex]\(8 \sqrt{10}\)[/tex], is already in its simplest form.
- For the second term, we need to simplify [tex]\(3 \sqrt{360}\)[/tex].
2. Simplify [tex]\(\sqrt{360}\)[/tex]:
- Begin by factoring 360 into its prime factors: [tex]\(360 = 36 \cdot 10 = 6^2 \cdot 10\)[/tex].
- Using the property of square roots, [tex]\(\sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b}\)[/tex], we get [tex]\(\sqrt{360} = \sqrt{36 \cdot 10} = \sqrt{36} \cdot \sqrt{10} = 6 \sqrt{10}\)[/tex].
3. Incorporate the Simplified Form of [tex]\(\sqrt{360}\)[/tex]:
- Substitute [tex]\(\sqrt{360}\)[/tex] with [tex]\(6 \sqrt{10}\)[/tex] in the original expression.
- The second term now becomes [tex]\(3 \sqrt{360} = 3(6 \sqrt{10}) = 18 \sqrt{10}\)[/tex].
4. Combine Like Terms:
- Now we have two terms with [tex]\(\sqrt{10}\)[/tex]: [tex]\(8 \sqrt{10} + 18 \sqrt{10}\)[/tex].
- Since both terms contain [tex]\(\sqrt{10}\)[/tex], we can add the coefficients together: [tex]\(8 + 18 = 26\)[/tex].
5. Write the Final Simplified Expression:
- Thus, the simplified form of the given expression is [tex]\(26 \sqrt{10}\)[/tex].
For the numerical values, we have:
- [tex]\( 8 \sqrt{10} \approx 25.298 \)[/tex]
- [tex]\( 3 \sqrt{360} \approx 56.921 \)[/tex]
- Combined, [tex]\( 8 \sqrt{10} + 3 \sqrt{360} \approx 82.219 \)[/tex]
So, the completely simplified expression in terms of the evaluation is:
[tex]\[ 8 \sqrt{10} + 3 \sqrt{360} \approx 82.219 \][/tex]
Therefore, these are the steps and the numerical result for the simplified form.
