To express [tex]\( \sqrt{99} + \sqrt{44} \)[/tex] in the form [tex]\( a \sqrt{b} \)[/tex] where [tex]\( a \)[/tex] and [tex]\( b \)[/tex] are integers, follow these steps:
1. Simplify [tex]\(\sqrt{99}\)[/tex]:
- Recognize that [tex]\( 99 \)[/tex] can be factored into [tex]\( 9 \times 11 \)[/tex].
- Thus, [tex]\( \sqrt{99} = \sqrt{9 \times 11} = \sqrt{9} \times \sqrt{11} \)[/tex].
- Since [tex]\( \sqrt{9} = 3 \)[/tex], we have [tex]\( \sqrt{99} = 3 \sqrt{11} \)[/tex].
2. Simplify [tex]\(\sqrt{44}\)[/tex]:
- Recognize that [tex]\( 44 \)[/tex] can be factored into [tex]\( 4 \times 11 \)[/tex].
- Thus, [tex]\( \sqrt{44} = \sqrt{4 \times 11} = \sqrt{4} \times \sqrt{11} \)[/tex].
- Since [tex]\( \sqrt{4} = 2 \)[/tex], we have [tex]\( \sqrt{44} = 2 \sqrt{11} \)[/tex].
3. Combine the simplified square roots:
- Adding the simplified forms, we get [tex]\( \sqrt{99} + \sqrt{44} = 3 \sqrt{11} + 2 \sqrt{11} \)[/tex].
- Combine the coefficients of [tex]\(\sqrt{11}\)[/tex]: [tex]\( 3 \sqrt{11} + 2 \sqrt{11} = (3 + 2) \sqrt{11} = 5 \sqrt{11} \)[/tex].
Therefore, the expression [tex]\( \sqrt{99} + \sqrt{44} \)[/tex] can be written in the form [tex]\( a \sqrt{b} \)[/tex] as [tex]\( 5 \sqrt{11} \)[/tex].
So, the values of [tex]\( a \)[/tex] and [tex]\( b \)[/tex] are:
[tex]\[ a = 5, \quad b = 11. \][/tex]
