Question

Rewrite the expression using the properties of exponents.


`( \frac{4 }{64 {}^{ (5)/(6) } } ) {}^{ (1)/(2) }`
​

Answer 1

`(\frac{4}{64^{(5)/(6)}})^{(1)/(2)}=(√(2))/(4)`

Explanation:

Exponents property :
`(x^a)^b=x^(ab);(x^c)/(x^a)=x^(c-d)=(1)/(x^(d-c))`

`(\frac{4}{64^{(5)/(6)}})^{(1)/(2)}\\4=2* 2=2^2\\64=2* 2* 2* 2* 2* 2=2^6\\\\64^{(5)/(6)}=(26)^{(5)/(6)}=2^{6* (5)/(6)}=2^5\\\\(\frac{4}{64^{(5)/(6)}})^{(1)/(2)}=((2^2)/(2^5))^{(1)/(2)}=((1)/(2^(5-2)))^{(1)/(2)}=((1)/(2^3))^{(1)/(2)}\\\\=\frac{1}{2^{(3)/(2)}}=(1)/(2√(2) )`

Multiply numerator and denominator by
`√(2)`

`(\frac{4}{64^{(5)/(6)}})^{(1)/(2)}=(1)/(2√(2))* (√(2))/(√(2))=(√(2))/(4)`