Simplify (3 square root of 12x^5)(4 square root of 12x^3)

$(3 \sqrt{12 x^{5}}) (4 \sqrt{12 x^{3}})$

Step 1

Rewrite $12 x^{5}$ as ${(2 x^{2})}^{2} \cdot (3 x)$ .

Factor $4$ out of $12$ .

$3 \sqrt{4 (3) x^{5}} (4 \sqrt{12 x^{3}})$

Rewrite $4$ as $2^{2}$ .

$3 \sqrt{2^{2} \cdot 3 x^{5}} (4 \sqrt{12 x^{3}})$

Factor out $x^{4}$ .

$3 \sqrt{2^{2} \cdot 3 (x^{4} x)} (4 \sqrt{12 x^{3}})$

Rewrite $x^{4}$ as ${(x^{2})}^{2}$ .

$3 \sqrt{2^{2} \cdot 3 ({(x^{2})}^{2} x)} (4 \sqrt{12 x^{3}})$

Move $3$ .

$3 \sqrt{2^{2} {(x^{2})}^{2} \cdot 3 x} (4 \sqrt{12 x^{3}})$

Rewrite $2^{2} {(x^{2})}^{2}$ as ${(2 x^{2})}^{2}$ .

$3 \sqrt{{(2 x^{2})}^{2} \cdot 3 x} (4 \sqrt{12 x^{3}})$

Add parentheses.

$3 \sqrt{{(2 x^{2})}^{2} \cdot (3 x)} (4 \sqrt{12 x^{3}})$

Step 2

Pull terms out from under the radical.

$3 (2 x^{2} \sqrt{3 x}) (4 \sqrt{12 x^{3}})$

Step 3

Multiply $2$ by $3$ .

$6 (x^{2} \sqrt{3 x}) (4 \sqrt{12 x^{3}})$

Step 4

Rewrite $12 x^{3}$ as ${(2 x)}^{2} \cdot (3 x)$ .

Factor $4$ out of $12$ .

$6 x^{2} \sqrt{3 x} (4 \sqrt{4 (3) x^{3}})$

Rewrite $4$ as $2^{2}$ .

$6 x^{2} \sqrt{3 x} (4 \sqrt{2^{2} \cdot 3 x^{3}})$

Factor out $x^{2}$ .

$6 x^{2} \sqrt{3 x} (4 \sqrt{2^{2} \cdot 3 (x^{2} x)})$

Move $3$ .

$6 x^{2} \sqrt{3 x} (4 \sqrt{2^{2} x^{2} \cdot 3 x})$

Rewrite $2^{2} x^{2}$ as ${(2 x)}^{2}$ .

$6 x^{2} \sqrt{3 x} (4 \sqrt{{(2 x)}^{2} \cdot 3 x})$

Add parentheses.

$6 x^{2} \sqrt{3 x} (4 \sqrt{{(2 x)}^{2} \cdot (3 x)})$

Step 5

Pull terms out from under the radical.

$6 x^{2} \sqrt{3 x} (4 (2 x \sqrt{3 x}))$

Step 6

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

Move $x$ .

$6 (x \cdot x^{2}) \sqrt{3 x} (4 (2 \sqrt{3 x}))$

Multiply $x$ by $x^{2}$ .

Raise $x$ to the power of $1$ .

$6 (x^{1} x^{2}) \sqrt{3 x} (4 (2 \sqrt{3 x}))$

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

$6 x^{1 + 2} \sqrt{3 x} (4 (2 \sqrt{3 x}))$

Add $1$ and $2$ .

$6 x^{3} \sqrt{3 x} (4 (2 \sqrt{3 x}))$

Step 7

Multiply $2$ by $4$ .

$6 x^{3} \sqrt{3 x} (8 \sqrt{3 x})$

Step 8

Multiply $6 x^{3} \sqrt{3 x} (8 \sqrt{3 x})$ .

Multiply $8$ by $6$ .

$48 x^{3} \sqrt{3 x} \sqrt{3 x}$

Raise $\sqrt{3 x}$ to the power of $1$ .

$48 x^{3} ({\sqrt{3 x}}^{1} \sqrt{3 x})$

Raise $\sqrt{3 x}$ to the power of $1$ .

$48 x^{3} ({\sqrt{3 x}}^{1} {\sqrt{3 x}}^{1})$

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

$48 x^{3} {\sqrt{3 x}}^{1 + 1}$

Add $1$ and $1$ .

$48 x^{3} {\sqrt{3 x}}^{2}$

Step 9

Rewrite ${\sqrt{3 x}}^{2}$ as $3 x$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3 x}$ as ${(3 x)}^{\frac{1}{2}}$ .

$48 x^{3} {({(3 x)}^{\frac{1}{2}})}^{2}$

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

$48 x^{3} {(3 x)}^{\frac{1}{2} \cdot 2}$

Combine $\frac{1}{2}$ and $2$ .

$48 x^{3} {(3 x)}^{\frac{2}{2}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$48 x^{3} {(3 x)}^{\frac{2}{2}}$

Rewrite the expression.

$48 x^{3} {(3 x)}^{1}$

Simplify.

$48 x^{3} (3 x)$

Step 10

Rewrite using the commutative property of multiplication.

$48 \cdot 3 x^{3} x$

Step 11

Multiply $x^{3}$ by $x$ by adding the exponents.

Move $x$ .

$48 \cdot 3 (x \cdot x^{3})$

Multiply $x$ by $x^{3}$ .

Raise $x$ to the power of $1$ .

$48 \cdot 3 (x^{1} x^{3})$

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

$48 \cdot 3 x^{1 + 3}$

Add $1$ and $3$ .

$48 \cdot 3 x^{4}$

Step 12

Multiply $48$ by $3$ .

$144 x^{4}$

Simplify (3 square root of 12x^5)(4 square root of 12x^3)

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