Simplify cube root of 7x^7* cube root of 6x^8

$\sqrt[3]{7 x^{7}} \cdot \sqrt[3]{6 x^{8}}$

Step 1

Rewrite $7 x^{7}$ as ${(x^{2})}^{3} \cdot (7 x)$ .

Factor out $x^{6}$ .

$\sqrt[3]{7 (x^{6} x)} \cdot \sqrt[3]{6 x^{8}}$

Rewrite $x^{6}$ as ${(x^{2})}^{3}$ .

$\sqrt[3]{7 ({(x^{2})}^{3} x)} \cdot \sqrt[3]{6 x^{8}}$

Reorder $7$ and ${(x^{2})}^{3}$ .

$\sqrt[3]{{(x^{2})}^{3} \cdot 7 x} \cdot \sqrt[3]{6 x^{8}}$

Add parentheses.

$\sqrt[3]{{(x^{2})}^{3} \cdot (7 x)} \cdot \sqrt[3]{6 x^{8}}$

Step 2

Pull terms out from under the radical.

$x^{2} \sqrt[3]{7 x} \cdot \sqrt[3]{6 x^{8}}$

Step 3

Rewrite $6 x^{8}$ as ${(x^{2})}^{3} \cdot (6 x^{2})$ .

Factor out $x^{6}$ .

$x^{2} \sqrt[3]{7 x} \cdot \sqrt[3]{6 (x^{6} x^{2})}$

Rewrite $x^{6}$ as ${(x^{2})}^{3}$ .

$x^{2} \sqrt[3]{7 x} \cdot \sqrt[3]{6 ({(x^{2})}^{3} x^{2})}$

Reorder $6$ and ${(x^{2})}^{3}$ .

$x^{2} \sqrt[3]{7 x} \cdot \sqrt[3]{{(x^{2})}^{3} \cdot 6 x^{2}}$

Add parentheses.

$x^{2} \sqrt[3]{7 x} \cdot \sqrt[3]{{(x^{2})}^{3} \cdot (6 x^{2})}$

Step 4

Pull terms out from under the radical.

$x^{2} \sqrt[3]{7 x} \cdot (x^{2} \sqrt[3]{6 x^{2}})$

Step 5

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

Move $x^{2}$ .

$x^{2} x^{2} \sqrt[3]{7 x} \cdot \sqrt[3]{6 x^{2}}$

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

$x^{2 + 2} \sqrt[3]{7 x} \cdot \sqrt[3]{6 x^{2}}$

Add $2$ and $2$ .

$x^{4} \sqrt[3]{7 x} \cdot \sqrt[3]{6 x^{2}}$

Step 6

Multiply $x^{4} \sqrt[3]{7 x} \sqrt[3]{6 x^{2}}$ .

Combine using the product rule for radicals.

$x^{4} \sqrt[3]{6 x^{2} (7 x)}$

Multiply $7$ by $6$ .

$x^{4} \sqrt[3]{42 x^{2} x}$

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

$x^{4} \sqrt[3]{42 (x^{1} x^{2})}$

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

$x^{4} \sqrt[3]{42 x^{1 + 2}}$

Add $1$ and $2$ .

$x^{4} \sqrt[3]{42 x^{3}}$

Step 7

Reorder $42$ and $x^{3}$ .

$x^{4} \sqrt[3]{x^{3} \cdot 42}$

Step 8

Pull terms out from under the radical.

$x^{4} (x \sqrt[3]{42})$

Step 9

Rewrite using the commutative property of multiplication.

$\sqrt[3]{42} x^{4} x$

Step 10

Multiply $x^{4}$ by $x$ by adding the exponents.

Move $x$ .

$\sqrt[3]{42} (x \cdot x^{4})$

Multiply $x$ by $x^{4}$ .

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

$\sqrt[3]{42} (x^{1} x^{4})$

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

$\sqrt[3]{42} x^{1 + 4}$

Add $1$ and $4$ .

$\sqrt[3]{42} x^{5}$

Simplify cube root of 7x^7* cube root of 6x^8

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