Question

Which expression is equivalent to ((4^((5)/(4)).4^((1)/(4)))/(4^((1)/(2))))^((1)/(2))?

Answer 1

Answer:  2

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 :

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

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

Hence, the correct answer $(\dfrac{4^{\frac{5}{4}}\cdot4^{\frac{1}{4}}}{4^{\frac{1}{2}}})^{\frac{1}{2}= 2$

Answer 2

Well, this is a good practice of indices.

All of the numbers involved have the same base.

so, for the inner bracket, the powers will be (division is a minus sign for the powers):
$\frac{5}{4}+\frac{1}{4}-\frac{1}{2}=1$

So the inner value is
$(4^{1})^{\frac{1}{2}}=4^{\frac{1}{2}} = 2$