Simplify 5 square root of 42x(4+4 square root of 7x)

$5 \sqrt{42 x} (4 + 4 \sqrt{7 x})$

Step 1

Apply the distributive property.

$5 \sqrt{42 x} \cdot 4 + 5 \sqrt{42 x} (4 \sqrt{7 x})$

Step 2

Multiply $4$ by $5$ .

$20 \sqrt{42 x} + 5 \sqrt{42 x} (4 \sqrt{7 x})$

Step 3

Multiply $5 \sqrt{42 x} (4 \sqrt{7 x})$ .

Multiply $4$ by $5$ .

$20 \sqrt{42 x} + 20 \sqrt{42 x} \sqrt{7 x}$

Combine using the product rule for radicals.

$20 \sqrt{42 x} + 20 \sqrt{7 x (42 x)}$

Multiply $42$ by $7$ .

$20 \sqrt{42 x} + 20 \sqrt{294 x \cdot x}$

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

$20 \sqrt{42 x} + 20 \sqrt{294 (x^{1} x)}$

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

$20 \sqrt{42 x} + 20 \sqrt{294 (x^{1} x^{1})}$

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

$20 \sqrt{42 x} + 20 \sqrt{294 x^{1 + 1}}$

Add $1$ and $1$ .

$20 \sqrt{42 x} + 20 \sqrt{294 x^{2}}$

Step 4

Simplify each term.

Rewrite $294 x^{2}$ as ${(7 x)}^{2} \cdot 6$ .

Factor $49$ out of $294$ .

$20 \sqrt{42 x} + 20 \sqrt{49 (6) x^{2}}$

Rewrite $49$ as $7^{2}$ .

$20 \sqrt{42 x} + 20 \sqrt{7^{2} \cdot 6 x^{2}}$

Move $6$ .

$20 \sqrt{42 x} + 20 \sqrt{7^{2} x^{2} \cdot 6}$

Rewrite $7^{2} x^{2}$ as ${(7 x)}^{2}$ .

$20 \sqrt{42 x} + 20 \sqrt{{(7 x)}^{2} \cdot 6}$

Pull terms out from under the radical.

$20 \sqrt{42 x} + 20 (7 x \sqrt{6})$

Multiply $7$ by $20$ .

$20 \sqrt{42 x} + 140 x \sqrt{6}$