Question

Simplify: (14^6*15^5)/(35^6*6^5)

Answer 1

Step-by-step explanation:

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Answer 2

$\displaystyle \sf \frac{ {14}^{6} \times {15}^{5} }{ {35}^{6} \times {6}^{5} } = \bf \frac{2}{5}$

Given :

$\displaystyle \sf \frac{ {14}^{6} \times {15}^{5} }{ {35}^{6} \times {6}^{5} }$

To find :

Simplify the given expression

Formula Used :

We are aware of the formula on indices that :

$\sf{1. \: \: { ({a}^{m} )}^{n} = {a}^{mn} }$

$\displaystyle \sf{2. \: \: {a}^{m} \times {a}^{n} = {a}^{m + n} }$

Solution :

Step 1 of 2 :

Write down the given expression

Here the given expression is

$\displaystyle \sf \frac{ {14}^{6} \times {15}^{5} }{ {35}^{6} \times {6}^{5} }$

Step 2 of 2 :

Simplify the given expression

$\displaystyle \sf \frac{ {14}^{6} \times {15}^{5} }{ {35}^{6} \times {6}^{5} }$

$\displaystyle \sf = \frac{ {14}^{6} }{ {35}^{6} } \times \frac{ {15}^{5} }{ {6}^{5} }$

$\displaystyle \sf = {\bigg( \frac{14}{35} \bigg)}^{6} \times {\bigg( \frac{15}{6} \bigg)}^{5}$

$\displaystyle \sf = {\bigg( \frac{2 \times 7}{5 \times 7} \bigg)}^{6} \times {\bigg( \frac{3 \times 5}{2 \times 3} \bigg)}^{5}$

$\displaystyle \sf = {\bigg( \frac{2 }{5 } \bigg)}^{6} \times {\bigg( \frac{ 5}{2 } \bigg)}^{5}$

$\displaystyle \sf = {\bigg( \frac{2 }{5 } \bigg)}^{6} \times {\bigg[{\bigg( \frac{2}{5} \bigg)}^{ - 1} \bigg] }^{5}$

$\displaystyle \sf = {\bigg( \frac{2 }{5 } \bigg)}^{6} \times {\bigg( \frac{2 }{5 } \bigg)}^{ - 5} \: \: \: \bigg[ \: \because \:{ ({a}^{m} )}^{n} = {a}^{mn} \bigg]$

$\displaystyle \sf = {\bigg( \frac{2 }{5 } \bigg)}^{6 - 5}\: \: \: \bigg[ \: \because \: {a}^{m} \times {a}^{n} = {a}^{m + n} \bigg]$

$\displaystyle \sf = {\bigg( \frac{2 }{5 } \bigg)}^{1}$

$\displaystyle \sf = \frac{2}{5}$

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