Mathematics: Which expression is equivalent to ((4^((5)/(((1)/(/(4^((1)/(((1)/(

Which expression is equivalent to ((4^((5)/(((1)/(/(4^((1)/(((1)/(

Ответ:

lizsd2004

06.01.2022

Mathematics

Step-by-step explanation:

The given expression : $(\dfrac{4^{\frac{5}{4}}\cdot4^{\frac{1}{4}}}{4^{\frac{1}{2}}})^{\frac{1}{2}$

Using product rule of exponents :

$a^m\cdot a^n= a^{m+n}$

we get

$(\dfrac{4^{\frac{5}{4}}\cdot4^{\frac{1}{4}}}{4^{\frac{1}{2}}})^{\frac{1}{2}}\\\\=(\dfrac{4^{\frac{5}{4}+\frac{1}{4}}}{4^{\frac{1}{2}}})^{\frac{1}{2}}\\\\=(\dfrac{4^{\frac{5+1}{4}}}{4^{\frac{1}{2}}})^{\frac{1}{2}}\\$

$=(\dfrac{4^{\frac{3}{2}}}{4^{\frac{1}{2}}})^{\frac{1}{2}}$

Using division rule of exponents :

$\dfrac{a^m}{a^n} =a^{m-n}$

$(\dfrac{4^{\frac{3}{2}}}{4^{\frac{1}{2}}})^{\frac{1}{2}}=(4^{\frac{3}{2}-\frac{1}{2}})^{\frac{1}{2}}\\\\=(4^{\frac{3-1}{2}})^{\frac{1}{2}}\\\\=(4^{\frac{2}{2}})^{\frac{1}{2}}\\\\=(4^1})^{\frac{1}{2}}=(2\times 2)^{\frac{1}{2}}= (2^2)^{\frac{1}{2}}=2$