Answer :
To simplify the expression [tex]\(\sqrt{40} + 2 \sqrt{10} + \sqrt{90}\)[/tex], we need to express each square root in terms of [tex]\(\sqrt{10}\)[/tex]. Let's go through it step-by-step:
1. Simplify [tex]\(\sqrt{40}\)[/tex]:
- Notice that [tex]\(40 = 4 \times 10\)[/tex].
- We can express [tex]\(\sqrt{40}\)[/tex] as [tex]\(\sqrt{4 \times 10} = \sqrt{4} \times \sqrt{10}\)[/tex].
- Since [tex]\(\sqrt{4} = 2\)[/tex], it simplifies to [tex]\(2\sqrt{10}\)[/tex].
2. Simplify [tex]\(\sqrt{90}\)[/tex]:
- Notice that [tex]\(90 = 9 \times 10\)[/tex].
- We can express [tex]\(\sqrt{90}\)[/tex] as [tex]\(\sqrt{9 \times 10} = \sqrt{9} \times \sqrt{10}\)[/tex].
- Since [tex]\(\sqrt{9} = 3\)[/tex], it simplifies to [tex]\(3\sqrt{10}\)[/tex].
3. Combine the simplified terms:
- Now substitute these simplifications into the original expression:
[tex]\[
\sqrt{40} + 2 \sqrt{10} + \sqrt{90} = 2\sqrt{10} + 2\sqrt{10} + 3\sqrt{10}
\][/tex]
4. Add all terms together:
- Add the coefficients of [tex]\(\sqrt{10}\)[/tex] together:
[tex]\[
2 + 2 + 3 = 7
\][/tex]
- Therefore, the entire expression simplifies to:
[tex]\[
7\sqrt{10}
\][/tex]
Thus, the choice that is equivalent to the expression [tex]\(\sqrt{40} + 2 \sqrt{10} + \sqrt{90}\)[/tex] is [tex]\( \boxed{7\sqrt{10}} \)[/tex]. So, the correct answer is B.
1. Simplify [tex]\(\sqrt{40}\)[/tex]:
- Notice that [tex]\(40 = 4 \times 10\)[/tex].
- We can express [tex]\(\sqrt{40}\)[/tex] as [tex]\(\sqrt{4 \times 10} = \sqrt{4} \times \sqrt{10}\)[/tex].
- Since [tex]\(\sqrt{4} = 2\)[/tex], it simplifies to [tex]\(2\sqrt{10}\)[/tex].
2. Simplify [tex]\(\sqrt{90}\)[/tex]:
- Notice that [tex]\(90 = 9 \times 10\)[/tex].
- We can express [tex]\(\sqrt{90}\)[/tex] as [tex]\(\sqrt{9 \times 10} = \sqrt{9} \times \sqrt{10}\)[/tex].
- Since [tex]\(\sqrt{9} = 3\)[/tex], it simplifies to [tex]\(3\sqrt{10}\)[/tex].
3. Combine the simplified terms:
- Now substitute these simplifications into the original expression:
[tex]\[
\sqrt{40} + 2 \sqrt{10} + \sqrt{90} = 2\sqrt{10} + 2\sqrt{10} + 3\sqrt{10}
\][/tex]
4. Add all terms together:
- Add the coefficients of [tex]\(\sqrt{10}\)[/tex] together:
[tex]\[
2 + 2 + 3 = 7
\][/tex]
- Therefore, the entire expression simplifies to:
[tex]\[
7\sqrt{10}
\][/tex]
Thus, the choice that is equivalent to the expression [tex]\(\sqrt{40} + 2 \sqrt{10} + \sqrt{90}\)[/tex] is [tex]\( \boxed{7\sqrt{10}} \)[/tex]. So, the correct answer is B.