Question

maths
solve and answer me​

Answer 1

3)

• Express the following in the form of p / q where p and q are integers and q ≠0 .

$\sf(a) \: \: 0. \bar{7} \: \: (b) \: \: 0. \overline{23} \: \: (c) \: \: 0.\overline{9} \: \: (d) \: \: 0.\overline{001}$

Solution

$\\\pink{ \sf(a) \: \: 0. \bar{7}}$

$Let \: x = 0. \overline{7}.\: \: Then$

${\implies \: x = 0.7777. .. .. . \: \: \: \: \: \: - - - - (1)}$

• Multiplying both sides by 10

${\implies \:10 x = 7.7777.. . . .. \: \: \: \: \: \: - - - - (2)}$

• On subtracting (1) From (2) we get

$\implies 10x - x = 7.7777. .. - 0.7777. .. .. .$

$\implies 9x = 7$

$\implies x = \dfrac{7}{9}$

$\pink{\implies \sf \: \: 0. \bar{7} = \dfrac{7}{9} }$

$\green{ \sf(b) \: \: 0. \overline{23}}$

$\implies let \: x = 0. \overline{23}$

${\implies \: x = 0.353535.. . \: \: \: \: \: \: \: - - - (1)}$

• Here we have to repeating decimal after the decimal point so we multiply side of (1 ) by 10²=100 to get

${\implies 100x = 23.23232... \: \: \: \: \: - - - (2)}$

• On subtracting (1) From (2) we get

$\implies 100x - x = (23.2323232. . .) - (0.232323.. .)$

$\implies 99x= 23$

$\green{x= \dfrac{23}{99}}$

$\blue{ (c) \: \: 0.\overline{9}}$

$Let \: x = 0. \overline{9}.\: \: Then$

${\implies \: x = 0.9999. .. . .. \: \: \: \: \: \: - - - - (1)}$

• Multiplying both sides by 10

${\implies \:10 x = 9.99999.. . .. \: \: \: \: \: \: - - - - (2)}$

• On subtracting (1) From (2) we get

$\implies 10x - x = 9.9999. .. - 0.9999. . .. .$

$\implies 9x = 9$

$\implies x = \dfrac{9}{9}$

$\blue{\sf \: \: 0. \bar{7} = 1 }$

$\orange{(d) \: \: 0.\overline{001}}$

$let \: x = 0. \overline{001001001. . .}$

${\implies \: x = 0.001001001 . . . \: \: \: \: \: \: \: - - - (1)}$

• Here we have to repeating decimal after the decimal point so we multiply side of (1 ) by 10³=1000 to get

${\implies1000x = 1.001001001 \: \: \: \: \: - - - (2)}$

• On subtracting (1) From (2) we get

$\implies100x - x = ( 1.001001001 ) -( 0.001001001. . .)$

$\implies999x= 1$

$\orange{\implies x= \dfrac{1}{999}}$

4)

simplify the following

$\sf\sqrt{48} - \sqrt{72} - \sqrt{27} + 2 \sqrt{18}$

Solution

${\sf \implies \: \sqrt{48} - \sqrt{72} - \sqrt{27} + 2 \sqrt{18} }$

${\sf \implies \: \sqrt{2 \times 2 \times 2 \times 2 \times 3} - \sqrt{2 \times 2 \times 2 \times 3 \times 3} - \sqrt{3 \times 3 \times 3} + 2 \sqrt{3 \times 3 \times 2} }$

${\sf \implies \: \sqrt{ ( {2}^{2} )^{2} \times 3} - \sqrt{ {2}^{3} \times {3}^{2} } - \sqrt{ {3}^{3} } + 2 \sqrt{ {3}^{2} \times 2} }$

${\sf \implies \:4 \sqrt{ 3} - \sqrt{ {2}^{2} \times {3}^{2} \times 2} - \sqrt{ {3}^{2} \times 3} + 2 \times 3 \sqrt{ 2} }$

${\sf \implies \:4 \sqrt{ 3} - \sqrt{ {6}^{2} \times 2} - 3\sqrt{ 3} + 6 \sqrt{ 2} }$

${\sf \implies \:4 \sqrt{ 3} - 3\sqrt{ 3} - 6\sqrt{ 2} + 6 \sqrt{ 2} }$

${\sf \implies \: \sqrt{3} \: \: \: \: \: \: \: \: \cancel{ - 6\sqrt{ 2} } \: \: \: \: \: \cancel{ + 6 \sqrt{ 2} }}$

${\sf \implies \: \sqrt{3} \: \: \: \: }$