which expression is equivalent to $left(\frac{4^{\frac{5}{4}}cdot4^{\frac{1}{4}}}{4^{\frac{1}{2}}}\right)^{\f…

which expression is equivalent to $left(\frac{4^{\frac{5}{4}}cdot4^{\frac{1}{4}}}{4^{\frac{1}{2}}}\right)^{\frac{1}{2}}$?\n$sqrt16{4^{5}}$\n$sqrt{2^{5}}$\n$2$\n$4$

which expression is equivalent to $left(\frac{4^{\frac{5}{4}}cdot4^{\frac{1}{4}}}{4^{\frac{1}{2}}}\right)^{\frac{1}{2}}$?\n$sqrt16{4^{5}}$\n$sqrt{2^{5}}$\n$2$\n$4$

Answer

Explanation:

Step1: Use exponent - product rule

According to the rule $a^m\cdot a^n=a^{m + n}$, for the numerator $4^{\frac{5}{4}}\cdot4^{\frac{1}{4}}$, we have $4^{\frac{5}{4}+\frac{1}{4}}=4^{\frac{5 + 1}{4}}=4^{\frac{6}{4}}=4^{\frac{3}{2}}$.

Step2: Rewrite the original expression

The original expression $\left(\frac{4^{\frac{5}{4}}\cdot4^{\frac{1}{4}}}{4^{\frac{1}{2}}}\right)^{\frac{1}{2}}$ becomes $\left(\frac{4^{\frac{3}{2}}}{4^{\frac{1}{2}}}\right)^{\frac{1}{2}}$.

Step3: Use exponent - quotient rule

According to the rule $\frac{a^m}{a^n}=a^{m - n}$, we get $\left(4^{\frac{3}{2}-\frac{1}{2}}\right)^{\frac{1}{2}}=\left(4^{\frac{3 - 1}{2}}\right)^{\frac{1}{2}}=(4^1)^{\frac{1}{2}}$.

Step4: Simplify the power - of - a - power

According to the rule $(a^m)^n=a^{mn}$, we have $4^{\frac{1}{2}}=\sqrt{4}=2$.

Answer:

2