Simplify (x(9x^2+6)^(1/2))^2

${(x {(9 x^{2} + 6)}^{\frac{1}{2}})}^{2}$

Step 1

Apply the product rule to $x {(9 x^{2} + 6)}^{\frac{1}{2}}$ .

$x^{2} {({(9 x^{2} + 6)}^{\frac{1}{2}})}^{2}$

Step 2

Multiply the exponents in ${({(9 x^{2} + 6)}^{\frac{1}{2}})}^{2}$ .

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

$x^{2} {(9 x^{2} + 6)}^{\frac{1}{2} \cdot 2}$

Cancel the common factor of $2$ .

Cancel the common factor.

$x^{2} {(9 x^{2} + 6)}^{\frac{1}{2} \cdot 2}$

Rewrite the expression.

$x^{2} {(9 x^{2} + 6)}^{1}$

Step 3

Simplify.

$x^{2} (9 x^{2} + 6)$

Step 4

Simplify by multiplying through.

Apply the distributive property.

$x^{2} (9 x^{2}) + x^{2} \cdot 6$

Reorder.

Rewrite using the commutative property of multiplication.

$9 x^{2} x^{2} + x^{2} \cdot 6$

Move $6$ to the left of $x^{2}$ .

$9 x^{2} x^{2} + 6 \cdot x^{2}$

Step 5

Multiply $x^{2}$ by $x^{2}$ by adding the exponents.

Move $x^{2}$ .

$9 (x^{2} x^{2}) + 6 \cdot x^{2}$

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

$9 x^{2 + 2} + 6 \cdot x^{2}$

Add $2$ and $2$ .

$9 x^{4} + 6 \cdot x^{2}$

$9 x^{4} + 6 x^{2}$