Simplify (0.25x^(1/4))^2(12x^3)

${(0.25 x^{\frac{1}{4}})}^{2} (12 x^{3})$

Step 1

Rewrite using the commutative property of multiplication.

$12 {(0.25 x^{\frac{1}{4}})}^{2} x^{3}$

Step 2

Apply the product rule to $0.25 x^{\frac{1}{4}}$ .

$12 ({0.25}^{2} {(x^{\frac{1}{4}})}^{2}) x^{3}$

Step 3

Raise $0.25$ to the power of $2$ .

$12 (0.0625 {(x^{\frac{1}{4}})}^{2}) x^{3}$

Step 4

Multiply the exponents in ${(x^{\frac{1}{4}})}^{2}$ .

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

$12 (0.0625 x^{\frac{1}{4} \cdot 2}) x^{3}$

Cancel the common factor of $2$ .

Factor $2$ out of $4$ .

$12 (0.0625 x^{\frac{1}{2 (2)} \cdot 2}) x^{3}$

Cancel the common factor.

$12 (0.0625 x^{\frac{1}{2 \cdot 2} \cdot 2}) x^{3}$

Rewrite the expression.

$12 (0.0625 x^{\frac{1}{2}}) x^{3}$

Step 5

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

Move $x^{3}$ .

$12 (0.0625 (x^{3} x^{\frac{1}{2}}))$

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

$12 (0.0625 x^{3 + \frac{1}{2}})$

To write $3$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$12 (0.0625 x^{3 \cdot \frac{2}{2} + \frac{1}{2}})$

Combine $3$ and $\frac{2}{2}$ .

$12 (0.0625 x^{\frac{3 \cdot 2}{2} + \frac{1}{2}})$

Combine the numerators over the common denominator.

$12 (0.0625 x^{\frac{3 \cdot 2 + 1}{2}})$

Simplify the numerator.

Multiply $3$ by $2$ .

$12 (0.0625 x^{\frac{6 + 1}{2}})$

Add $6$ and $1$ .

$12 (0.0625 x^{\frac{7}{2}})$

Step 6

Multiply $0.0625$ by $12$ .

$0.75 x^{\frac{7}{2}}$