Question

please solve this problems and send me the answers with solutions . solve all the Questions. please ​

Answer 1

We know,

• Using the following identities, we can solve the problems vased on exponent & powers :

$\rm {a}^{m} \times {a}^{n} = {a}^{( m+n )} \\ \rm {a}^{m} \div {a}^{n} = {a}^{( m - n )} \\ \rm {a}^{n} \times {b}^{n} = {(a \times b)}^{n} \\ \rm \: \: \: \: \: \: \: \: \: \: {a}^{0} = 1$

Now, Solving the Problems ::

11.] Evaluate :

$\rm{}(i) \: \: \sf ( {6}^{0} + {15}^{0} ) \times ( {12}^{0} + 5) \\ \sf \to \: (1 + 1) \times (1 +5 ) \\ \sf\to \: (2) \times (6) \: \: \: = \red{12}$

$\rm(ii) \: \: \sf \frac{2 \times {3}^{3} \times 64 }{81 \times {4}^{2} } \\ \\ \sf \to \: \: \frac{2 \times {3}^{3} \times {2}^{6} }{ {3}^{4} \times {( {2}^{2}) }^{2} } \: = \frac{ {2}^{(1 + 6)} \times {3}^{3} }{ {3}^{4} \times {2}^{4} } \\ \\ \sf \to \frac{ {2}^{7} \times {3}^{3} }{ {3}^{4} \times {2}^{4} } \: \: = {2}^{(7 - 4)} \times {3}^{(3 - 4)} \\ \\ \sf \to {2}^{3} \times {3}^{ - 1} \: \: = {2}^{3} \times \frac{1}{ {(3)}^{1} } \\ \\ \sf \to \frac{ {2}^{3} }{3} \: \: \: \: = \red{ \frac{8}{3} }$

$\rm{} \: (iii) \: \: \sf\frac{ {10}^{5} \times 125 \times {3}^{5} }{ {5}^{7} \times {6}^{4} } \\ \\ \sf \to \frac{ {(2 \times 5)}^{5} \times {5}^{3} \times {3}^{5} }{ {5}^{7} \times {(2 \times 3)}^{4} } \\ \\ \sf\to \frac{ {2}^{5} \times {5}^{5} \times {5}^{3} \times {3}^{5} }{ {5}^{7} \times {2}^{4} \times {3}^{4} } \\ \\ \sf \to \frac{ {2}^{5} \times {5}^{(5 + 3)} \times {3}^{5} }{{5}^{7} \times {2}^{4} \times {3}^{4}} \: = \frac{ {2}^{5} \times {5}^{8} \times {3}^{5} }{ {5}^{7} \times {2}^{4} \times {3}^{4} } \\ \\ \sf \to {2}^{(5 - 4)} \times {5}^{(8 - 7)} \times {3}^{(5 - 4)} \\ \sf \to \: {2}^{1} \times {5}^{1} \times {3}^{1} \: \: = 2 \times 5 \times 3 = \red{30}$

12.] Simplify :

$\sf(i) \: \: {2}^{55} \times {2}^{60} - {2}^{97} \times {2}^{18} \\ \sf \to \: {2}^{(55 + 60)} - {2}^{(97 + 18)} \\ \sf \to \: {2}^{115} - {2}^{115} = \red0$

$\sf(ii) \: \: {2}^{3} \times {a}^{3} \times {5a}^{4} \\ \sf \to {(2 \times a)}^{3} \times {5a}^{ 4 } \\ \sf \to {(2a)}^{3} \times {5a}^{4} \\ \sf \to \: {8a}^{3} \times {5a}^{4} = (8 \times5 ) {a}^{(3 + 4)} \\ \sf \to \: {40a}^{7}$

$\sf (iii) \: \: \frac{ {3}^{n} + {3}^{n + 1} }{ {3}^{n + 1} - {3}^{n} } \\ \\ \sf \to \frac{ {3}^{n} + {3}^{n} \times {3}^{1} }{{3}^{n} \times {3}^{1} - {3}^{n} } \\ \\ \sf \to \frac{ {3}^{n} (1 + {3}^{1} )}{ {3}^{n} ( {3}^{1} - 1)} = \frac{ {3}^{n} (1 + 3)}{ {3}^{n} (3 - 1)} \\ \\ \sf \to \frac{ \cancel{{3}^{n}}(4) }{ \cancel{{3}^{n} }(2)} \: \: = \frac{4}{2} \: \: \: = \red2$

Answer 2

Answer:

hope this answers will help u.