Simplify (2^(1/2)*2^(3/4))^2

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

Step 1

Multiply $2^{\frac{1}{2}}$ by $2^{\frac{3}{4}}$ by adding the exponents.

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

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

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

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

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

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

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

Multiply $2$ by $2$ .

${(2^{\frac{2}{4} + \frac{3}{4}})}^{2}$

Combine the numerators over the common denominator.

${(2^{\frac{2 + 3}{4}})}^{2}$

Add $2$ and $3$ .

${(2^{\frac{5}{4}})}^{2}$

Step 2

Multiply the exponents in ${(2^{\frac{5}{4}})}^{2}$ .

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

$2^{\frac{5}{4} \cdot 2}$

Cancel the common factor of $2$ .

Factor $2$ out of $4$ .

$2^{\frac{5}{2 (2)} \cdot 2}$

Cancel the common factor.

$2^{\frac{5}{2 \cdot 2} \cdot 2}$

Rewrite the expression.

$2^{\frac{5}{2}}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$2^{\frac{5}{2}}$

Decimal Form:

$5.65685424 \dots$