Question

simplify the following : ​

Answer 1

$\huge \fbox \orange{Solutions:}$

$1. \: \sqrt{45} - 3 \sqrt{20} + 4√5$

$\sf \colorbox{pink} {Answer. 1}$

$\sqrt{45} - 3 \sqrt{20} + 4√5$

$= \sqrt{3 \times 3 \times 5} - 3 \sqrt{2 \times 2 \times 5} + 4 \sqrt{5}$

$= 3 \sqrt{5} - 3 \times 2 \sqrt{5} + 4 \sqrt{5}$

$= 3 \sqrt{5} - 6 \sqrt{5} + 4 \sqrt{5}$

$= 7 \sqrt{5} - 6 \sqrt{5}$

$= \sqrt{5}$

$2. \: \frac{ \sqrt{24} }{8} + \frac{ \sqrt{54} }{9}$

$\sf \colorbox{pink} {Answer. 2}$

$\textbf{Given:} \frac{ \sqrt{24} }{8} + \frac{ \sqrt{54} }{9}$

Solution:

$= \frac{ \sqrt{24} }{8} + \frac{ \sqrt{54} }{9}$

$= \frac{ \sqrt{2 \times 2 \times 2 \times 3} }{8} + \frac{ \sqrt{2 \times 3 \times 3} }{9}$

$⇒ \frac{2 \sqrt{2 \times 3} }{8} + \frac{3 \sqrt{2 \times 3} }{9} = \frac{ \sqrt{6} }{4} + \frac{ \sqrt{6} }{3}$

$⇒ \sqrt{6} \left( \frac{1}{4} + \frac{1}{3} \right) \] = \sqrt{6} \times \frac{7}{12}$

$= \frac{7}{12} \sqrt{6}$

$3. \: \: \sqrt{3} - 2 \sqrt{27} + \frac{7}{ \sqrt{3} }$

$\sf \colorbox{pink} {Answer. 3}$

$\textbf{Given:} \sqrt{3} - 2 \sqrt{27} + \frac{7}{ \sqrt{3} }$

Solution:

$⇒3 \sqrt{3} - 2 \sqrt{27} + \frac{7}{ \sqrt{3} }$

$⇒3 \sqrt{3 } - 6 \sqrt{3} + \frac{7}{ \sqrt{3} }$

$⇒9 \sqrt{3} - \frac{7}{ \sqrt{3} } = \left( \frac{9 \sqrt{3} }{ \sqrt{3} } \times \sqrt{3} \right) \] + \frac{7}{3}$

$⇒ \frac{27}{ \sqrt{3} } + \frac{7}{ \sqrt{3} } = \frac{34}{ \sqrt{3} }$

$4. \: \sqrt[4]{81} - 8 \sqrt[3]{216} + 15 \sqrt[5]{32} + \sqrt{225}$

$\sf \colorbox{pink} {Answer .4 }$

$\textbf{Given:} \sqrt[4]{81} - 8 \sqrt[3]{216} + 15 \sqrt[5]{32} + \sqrt{225}$

Solution:

$= \sqrt[4]{ {3}^{4} } - 8$

$= \sqrt[3]{ {6}^{3} } + 15$

$= \sqrt[5]{ {2}^{5} } + \sqrt{ {15}^{2} }$

$= 3 - 8 \times 6 + 15 \times 2 + 15$

$= 3 - 48 + 30 + 15 = 0$

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$\\ \\ \\ \\ \\ \\ \\ \sf \colorbox{gold} {\red(ANSWER ᵇʸ ⁿᵃʷᵃᵇ⁰⁰⁰⁸}$