Question

Please help with all of the questions, do not spam.

Subject: Mathematics.

Answer 1

➤ Answer (1) :-

${\tt \longrightarrow \dfrac{{2}^{3} \times {3}^{4} \times 4}{3 \times 32}}$

Convert all of them into exponents and powers form.

${\tt \longrightarrow \dfrac{{2}^{3} \times {3}^{4} \times {2}^{2}}{{3}^{1} \times {2}^{5}}}$

Simplify each of them...

${\tt \longrightarrow \dfrac{{2}^{3 + 2} \times {3}^{4}}{{3}^{1} \times {2}^{5}} = \dfrac{{2}^{5} \times {3}^{4}}{{3}^{1} \times {2}^{5}}}$

${\tt \longrightarrow {2}^{5 - 5} \times {3}^{4 - 1} = {2}^{0} \times {3}^{3}}$

${\tt \longrightarrow {3}^{3}}$

➤ Answer (2) :-

${\tt \longrightarrow \bigg(( {5}^{2} {)}^{3} \times {5}^{4} \bigg) \div {5}^{7}}$

${\tt \longrightarrow \bigg({5}^{2 \times 3}\times {5}^{4} \bigg) \div {5}^{7}}$

${\tt \longrightarrow {5}^{6 + 4} \div {5}^{7} = {5}^{10} \div {5}^{7}}$

${\tt \longrightarrow {5}^{10 - 7}}$

${\tt \longrightarrow {5}^{3}}$

➤ Answer (3) :-

${\tt \longrightarrow {25}^{4} \times {5}^{3}}$

${\tt \longrightarrow ( {5}^{2})^{4} \times {5}^{3} = {5}^{2 \times 4} \times {5}^{3}}$

${\tt \longrightarrow {5}^{8} \times {5}^{3} = {5}^{8 + 3}}$

${\tt \longrightarrow {5}^{11}}$

➤ Answer (4) :-

$\tt \longrightarrow \dfrac{3 \times {7}^{2} \times {11}^{8}}{21 \times {11}^{3}}$

$\tt \longrightarrow \dfrac{{3}^{1} \times {7}^{2} \times {11}^{8}}{ {7}^{1} \times {3}^{1} \times {11}^{3}}$

${\tt \longrightarrow {3}^{1 - 1} \times {7}^{2 - 1} \times {11}^{8 - 3}}$

${\tt \longrightarrow {3}^{0} \times {7}^{1} \times {11}^{5}}$

${\tt \longrightarrow {7}^{1} \times {11}^{5}}$

➤ Answer (5) :-

$\tt \longrightarrow \dfrac{{3}^{7}}{ {3}^{4} \times {3}^{3}}$

$\tt \longrightarrow \dfrac{{3}^{7}}{ {3}^{4 + 3}} = \dfrac{{3}^{7}}{{3}^{7}}$

$\tt \longrightarrow {3}^{7 - 7}$

$\tt \longrightarrow {3}^{0}$

➤ Answer (6) :-

${\tt \longrightarrow {2}^{0} + {3}^{0} + {4}^{0}}$

${\tt \longrightarrow 1 + 1 + 1 = 3}$

${\tt \longrightarrow {3}^{1}}$

➤ Answer (7) :-

${\tt \longrightarrow {2}^{0} \times {3}^{0} \times {4}^{0}}$

${\tt \longrightarrow 1 \times 1 \times 1 = 1}$

${\tt \longrightarrow {1}^{1}}$

➤ Answer (8) :-

${\tt \longrightarrow ({3}^{0} + {2}^{0}) \times {5}^{0}}$

${\tt \longrightarrow (1 + 1) \times 1 = 2 \times 1}$

${\tt \longrightarrow {2}^{1}}$

➤ Answer (9) :-

$\tt \longrightarrow \dfrac{{2}^{8} \times {a}^{5}}{{4}^{3} \times {a}^{3}}$

$\tt \longrightarrow \dfrac{{2}^{8} \times {a}^{5}}{( {2}^{2}{)}^{3} \times {a}^{3}}$

$\tt \longrightarrow \dfrac{{2}^{8} \times {a}^{5}}{{2}^{2 \times 3} \times {a}^{3}} = \dfrac{{2}^{8} \times {a}^{5}}{{2}^{6} \times {a}^{3}}$

$\tt \longrightarrow {2}^{8 - 6} \times {a}^{5 - 3}$

$\tt \longrightarrow {2}^{2} \times {a}^{2}$

➤ Answer (10) :-

${\tt \longrightarrow \bigg(\dfrac{{a}^{5}}{{a}^{3}} \bigg) \times {a}^{8}}$

${\tt \longrightarrow {a}^{5 - 3} \times {a}^{8}}$

${\tt \longrightarrow {a}^{2} \times {a}^{8} = {a}^{2 + 8}}$

${\tt \longrightarrow {a}^{10}}$

➤ Answer (11) :-

${\tt \longrightarrow \dfrac{{4}^{5} \times {a}^{8} \: {b}^{3}}{{4}^{5} \times {a}^{5} \: {b}^{2}}}$

${\tt \longrightarrow {4}^{5 - 5} \times {a}^{8 - 5} \times {b}^{3 - 2}}$

${\tt \longrightarrow {4}^{0} \times {a}^{3} \times {b}^{1}}$

${\tt \longrightarrow {a}^{3} \times {b}^{1}}$

➤ Answer (12) :-

${\tt \longrightarrow \bigg( {2}^{3} \times 2 \bigg)^{2}}$

${\tt \longrightarrow \bigg( {2}^{3 + 1}\bigg)^{2} = {2}^{4 \times 2}}$

${\tt \longrightarrow {2}^{8}}$

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$\dashrightarrow$ Some related equations :-

${\to \sf {a}^{m} \times {a}^{n} = {a}^{m + n}}$

${\to \sf {a}^{m} \div {a}^{n} = {a}^{m - n}}$

${\to \sf \bigg( {a}^{m} \bigg)^{n} = {a}^{m \times n}}$

${\to \sf \dfrac{ {a}^{m}}{ {b}^{m}} = \bigg( {\dfrac{a}{b}}\bigg)^{m}}$

${\to \sf {a}^{0} = 1}$${\to \sf {a}^{ - 1} = \dfrac{1}{a}}$