Question

please help no silly answer needed​

Answer 1

Answer:

10

Step-by-step explanation:

Used formulae:-

→ (ab)^m = a^m × b^m

→ (a/b)^m = a^m / b^m

→ a^m × a^n = a^(m+n)

→ a^m / a^n = a^(m-n)

→ (a^m)^n = a^(mn)

→ a^0 = 1

→ a^-n = 1/(a^n)

Answer 2

Question :-

Find the value of

$\rm \: \dfrac{{\bigg(2\bigg) }^{\dfrac{1}{2}} \times \sqrt[3]{3} \times \sqrt[4]{4} }{ \sqrt[ - 5]{10} \times {\bigg(5\bigg) }^{\dfrac{3}{5}}} \div \dfrac{{\bigg(3\bigg) }^{\dfrac{4}{3}} \times {\bigg(5\bigg) }^{ - \dfrac{7}{5} }}{{\bigg(4\bigg) }^{ - \dfrac{3}{5}} \times 6}$

$\large\underline{\sf{Solution-}}$

Given expression is

$\rm \: \dfrac{{\bigg(2\bigg) }^{\dfrac{1}{2}} \times \sqrt[3]{3} \times \sqrt[4]{4} }{ \sqrt[ - 5]{10} \times {\bigg(5\bigg) }^{\dfrac{3}{5}}} \div \dfrac{{\bigg(3\bigg) }^{\dfrac{4}{3}} \times {\bigg(5\bigg) }^{ - \dfrac{7}{5} }}{{\bigg(4\bigg) }^{ - \dfrac{3}{5}} \times 6}$

We know,

$\boxed{\sf{ \: \: \sqrt[n]{a} \: = \: {\bigg(a\bigg) }^{\dfrac{1}{n} } \: }}$

So, using this result, can be rewritten as

$\rm \: \dfrac{{\bigg(2\bigg) }^{\dfrac{1}{2}} \times {\bigg(3\bigg) }^{\dfrac{1}{3}} \times {\bigg(4\bigg) }^{\dfrac{1}{4} }}{{\bigg(10\bigg) }^{ - \dfrac{1}{5} } \times {\bigg(5\bigg) }^{\dfrac{3}{5}}} \div \dfrac{{\bigg(3\bigg) }^{\dfrac{4}{3}} \times {\bigg(5\bigg) }^{ - \dfrac{7}{5} }}{{\bigg(4\bigg) }^{ - \dfrac{3}{5}} \times 3 \times 2}$

$\rm \: \dfrac{{\bigg(2\bigg) }^{\dfrac{1}{2}} \times {\bigg(3\bigg) }^{\dfrac{1}{3}} \times {\bigg(2\bigg) }^{\dfrac{2}{4} }}{{\bigg(2 \times 5\bigg) }^{ - \dfrac{1}{5} } \times {\bigg(5\bigg) }^{\dfrac{3}{5}}} \div \dfrac{{\bigg(3\bigg) }^{\dfrac{4}{3}} \times {\bigg(5\bigg) }^{ - \dfrac{7}{5} }}{{\bigg(2\bigg) }^{ - \dfrac{6}{5}} \times 3 \times 2}$

We know,

$\boxed{\tt{ \: \: {(x \times y)}^{n} \: = \: {x}^{n} \times {y}^{n} \: \: }}$

So, using this identity, we get

$\rm \: \dfrac{{\bigg(2\bigg) }^{\dfrac{1}{2}} \times {\bigg(3\bigg) }^{\dfrac{1}{3}} \times {\bigg(2\bigg) }^{\dfrac{2}{4} }}{{\bigg(2\bigg) }^{ - \dfrac{1}{5} } \times {\bigg(5\bigg) }^{ - \dfrac{1}{5} } \times {\bigg(5\bigg) }^{\dfrac{3}{5}}} \times \dfrac{{\bigg(2\bigg) }^{ - \dfrac{6}{5}} \times 3 \times 2}{{\bigg(3\bigg) }^{\dfrac{4}{3}} \times {\bigg(5\bigg) }^{ - \dfrac{7}{5} }}$

We know

$\boxed{\sf{ \: \: {x}^{m} \times {x}^{n} \: = \: {x}^{m + n} \: }}$

$\boxed{\sf{ \: \: {x}^{m} \div {x}^{n} \: = \: {x}^{m - n} \: }}$

So, using these Identities, we get

$\rm \: = \: {\bigg(2\bigg) }^{\dfrac{1}{2} + \dfrac{1}{2} + \dfrac{1}{5} - \dfrac{6}{5} + 1} \times {\bigg(3\bigg) }^{\dfrac{1}{3} + 1 - \dfrac{4}{3} } \times {\bigg(5\bigg) }^{\dfrac{1}{5} + \dfrac{7}{5} - \dfrac{3}{5} }$

$\rm \: = \: {\bigg(2\bigg) }^{\dfrac{5 + 5 + 2 - 12 + 10}{10}} \times {\bigg(3\bigg) }^{\dfrac{1 + 3 - 4}{3}} \times {\bigg(5\bigg) }^{\dfrac{1 + 7 - 3}{5} }$

$\rm \: = \: {\bigg(2\bigg) }^{\dfrac{10}{10}} \times {\bigg(3\bigg) }^{\dfrac{0}{3}} \times {\bigg(5\bigg) }^{\dfrac{5}{5} }$

$\rm \: = \: {2}^{1} \times {3}^{0} \times {5}^{1}$

$\rm \: = \: 2 \times 1 \times 5$

$\rm \: = \: 10$

Hence,

$\boxed{\tt{ \rm \: \dfrac{{\bigg(2\bigg) }^{\dfrac{1}{2}} \times \sqrt[3]{3} \times \sqrt[4]{4} }{ \sqrt[ - 5]{10} \times {\bigg(5\bigg) }^{\dfrac{3}{5}}} \div \dfrac{{\bigg(3\bigg) }^{\dfrac{4}{3}} \times {\bigg(5\bigg) }^{ - \dfrac{7}{5} }}{{\bigg(4\bigg) }^{ - \dfrac{3}{5}} \times 6} = 10 \:}}$

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ADDITIONAL INFORMATION

$\boxed{\tt{ \: {x}^{ - n} \: = \: \frac{1}{ {x}^{n} } \: \: }}$

$\boxed{\tt{ \: \: {\bigg[\dfrac{x}{y} \bigg]}^{ - n} = {\bigg[\dfrac{y}{x} \bigg]}^{n} \: \: }}$

$\boxed{\tt{ \: \: {x}^{0} \: = \: 1 \: \: }}$