Question

simplify (64)-2/3
____
(125)​

Answer 1

25/16

Step-by-step explanation:

(64/125)-⅔

(125/64)³/²

(5/4)³/²*³

(5/4)²

25/16

Answer 2

ɢɪᴠᴇɴ :

${( \frac{64}{125}) }^{ \frac{-2}{3} }$

ᴛᴏ ꜰɪɴᴅ :

Simplified form

ꜱᴏʟᴜᴛɪᴏɴ:

${( \frac{64}{125}) }^{ \frac{-2}{3} }$

We know that,

${( \frac{a}{b} )}^{m} = \frac{ {a}^{m} }{ {b}^{m} }$

$\frac{ {(64)}^{ \frac{-2}{3} } }{ {(125)}^{ \frac{-2}{3} } }$

By taking prime factorisation of 64 and 125

$=\frac{ {(4 \times 4 \times 4)}^{ \frac{-2}{3} } }{ {(5 \times 5 \times 5)}^{ \frac{-2}{3} } } \\ \\ = \frac{ {( {4}^{3} )}^{ \frac{-2}{3} } }{ {( {5}^{3} )}^{ \frac{-2}{3} } } \: \: \: \: \: \: \: \: \: \: \: \: \:$

We know that,

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

$= \frac{ { {4}^{3 \times }}^{ \frac{-2}{3} } }{ { {5}^{3 \times } }^{ \frac{-2}{3} } } \\ \\ = \frac{ {4}^{-2} }{ {5}^{-2} } \: \: \: \: \: \\ \\ We\: \: know\: \: that ,\\{ a }^{-m}= \frac{1}{{a}^{m}} \\ \\= \frac{1/16}{1/25} \\ \\=\frac{25}{16} \: \: \: \: \:$