Write the Fraction in Simplest Form ((4^(5/4)*0.4^(1/4))/(4^(1/2)))^(1/2)

${(\frac{4^{\frac{5}{4}} \cdot {0.4}^{\frac{1}{4}}}{4^{\frac{1}{2}}})}^{\frac{1}{2}}$

Step 1

Move $4^{\frac{1}{2}}$ to the numerator using the negative exponent rule $\frac{1}{b^{n}} = b^{- n}$ .

${(4^{\frac{5}{4}} \cdot {0.4}^{\frac{1}{4}} \cdot 4^{- \frac{1}{2}})}^{\frac{1}{2}}$

Step 2

Multiply $4^{\frac{5}{4}}$ by $4^{- \frac{1}{2}}$ by adding the exponents.

Move $4^{- \frac{1}{2}}$ .

${(4^{- \frac{1}{2}} \cdot 4^{\frac{5}{4}} \cdot {0.4}^{\frac{1}{4}})}^{\frac{1}{2}}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

${(4^{- \frac{1}{2} + \frac{5}{4}} \cdot {0.4}^{\frac{1}{4}})}^{\frac{1}{2}}$

To write $- \frac{1}{2}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

${(4^{- \frac{1}{2} \cdot \frac{2}{2} + \frac{5}{4}} \cdot {0.4}^{\frac{1}{4}})}^{\frac{1}{2}}$

Write each expression with a common denominator of $4$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{1}{2}$ by $\frac{2}{2}$ .

${(4^{- \frac{2}{2 \cdot 2} + \frac{5}{4}} \cdot {0.4}^{\frac{1}{4}})}^{\frac{1}{2}}$

Multiply $2$ by $2$ .

${(4^{- \frac{2}{4} + \frac{5}{4}} \cdot {0.4}^{\frac{1}{4}})}^{\frac{1}{2}}$

Combine the numerators over the common denominator.

${(4^{\frac{- 2 + 5}{4}} \cdot {0.4}^{\frac{1}{4}})}^{\frac{1}{2}}$

Add $- 2$ and $5$ .

${(4^{\frac{3}{4}} \cdot {0.4}^{\frac{1}{4}})}^{\frac{1}{2}}$

Step 3

Raise $0.4$ to the power of $\frac{1}{4}$ .

${(4^{\frac{3}{4}} \cdot 0.79527072)}^{\frac{1}{2}}$

Step 4

Multiply $4^{\frac{3}{4}}$ by $0.79527072$ .

${2.2493653}^{\frac{1}{2}}$

Step 5

Rewrite $2.2493653$ as ${1.49978841}^{2}$ .

${({1.49978841}^{2})}^{\frac{1}{2}}$

Step 6

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

${1.49978841}^{2 (\frac{1}{2})}$

Step 7

Cancel the common factor of $2$ .

Cancel the common factor.

${1.49978841}^{2 (\frac{1}{2})}$

Rewrite the expression.

${1.49978841}^{1}$

Step 8

Evaluate the exponent.

$1.49978841$