Simplify square root of 7x^5* square root of 42xy^9

$\sqrt{7 x^{5}} \cdot \sqrt{42 x y^{9}}$

Step 1

Rewrite $7 x^{5}$ as ${(x^{2})}^{2} \cdot (7 x)$ .

Factor out $x^{4}$ .

$\sqrt{7 (x^{4} x)} \cdot \sqrt{42 x y^{9}}$

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

$\sqrt{7 ({(x^{2})}^{2} x)} \cdot \sqrt{42 x y^{9}}$

Reorder $7$ and ${(x^{2})}^{2}$ .

$\sqrt{{(x^{2})}^{2} \cdot 7 x} \cdot \sqrt{42 x y^{9}}$

Add parentheses.

$\sqrt{{(x^{2})}^{2} \cdot (7 x)} \cdot \sqrt{42 x y^{9}}$

Step 2

Pull terms out from under the radical.

$x^{2} \sqrt{7 x} \cdot \sqrt{42 x y^{9}}$

Step 3

Rewrite $42 x y^{9}$ as ${({(y^{2})}^{2})}^{2} \cdot (42 x y)$ .

Factor out $y^{8}$ .

$x^{2} \sqrt{7 x} \cdot \sqrt{42 x (y^{8} y)}$

Rewrite $y^{8}$ as ${(y^{4})}^{2}$ .

$x^{2} \sqrt{7 x} \cdot \sqrt{42 x ({(y^{4})}^{2} y)}$

Move $x$ .

$x^{2} \sqrt{7 x} \cdot \sqrt{42 {(y^{4})}^{2} x y}$

Reorder $42$ and ${(y^{4})}^{2}$ .

$x^{2} \sqrt{7 x} \cdot \sqrt{{(y^{4})}^{2} \cdot 42 x y}$

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

$x^{2} \sqrt{7 x} \cdot \sqrt{{({(y^{2})}^{2})}^{2} \cdot 42 x y}$

Add parentheses.

$x^{2} \sqrt{7 x} \cdot \sqrt{{({(y^{2})}^{2})}^{2} \cdot 42 (x y)}$

Add parentheses.

$x^{2} \sqrt{7 x} \cdot \sqrt{{({(y^{2})}^{2})}^{2} \cdot (42 x y)}$

Step 4

Pull terms out from under the radical.

$x^{2} \sqrt{7 x} \cdot ({(y^{2})}^{2} \sqrt{42 x y})$

Step 5

Multiply the exponents in ${(y^{2})}^{2}$ .

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

$x^{2} \sqrt{7 x} \cdot (y^{2 \cdot 2} \sqrt{42 x y})$

Multiply $2$ by $2$ .

$x^{2} \sqrt{7 x} \cdot (y^{4} \sqrt{42 x y})$

Step 6

Multiply $x^{2} \sqrt{7 x} (y^{4} \sqrt{42 x y})$ .

Combine using the product rule for radicals.

$x^{2} (y^{4} \sqrt{7 x (42 x y)})$

Multiply $42$ by $7$ .

$x^{2} (y^{4} \sqrt{294 x (x y)})$

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

$x^{2} (y^{4} \sqrt{294 (x^{1} x) y})$

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

$x^{2} (y^{4} \sqrt{294 (x^{1} x^{1}) y})$

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

$x^{2} (y^{4} \sqrt{294 x^{1 + 1} y})$

Add $1$ and $1$ .

$x^{2} (y^{4} \sqrt{294 x^{2} y})$

Step 7

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

Factor $49$ out of $294$ .

$x^{2} (y^{4} \sqrt{49 (6) x^{2} y})$

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

$x^{2} (y^{4} \sqrt{7^{2} \cdot 6 x^{2} y})$

Move $6$ .

$x^{2} (y^{4} \sqrt{7^{2} x^{2} \cdot 6 y})$

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

$x^{2} (y^{4} \sqrt{{(7 x)}^{2} \cdot 6 y})$

Add parentheses.

$x^{2} (y^{4} \sqrt{{(7 x)}^{2} \cdot (6 y)})$

Step 8

Pull terms out from under the radical.

$x^{2} (y^{4} (7 x \sqrt{6 y}))$

Step 9

Rewrite using the commutative property of multiplication.

$7 x^{2} (y^{4} x \sqrt{6 y})$

Step 10

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

Move $x$ .

$7 (x \cdot x^{2}) (y^{4} \sqrt{6 y})$

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

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

$7 (x^{1} x^{2}) (y^{4} \sqrt{6 y})$

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

$7 x^{1 + 2} (y^{4} \sqrt{6 y})$

Add $1$ and $2$ .

$7 x^{3} (y^{4} \sqrt{6 y})$

$7 x^{3} y^{4} \sqrt{6 y}$

Simplify square root of 7x^5* square root of 42xy^9

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