Finite Math Examples

$\sqrt[5]{x^{3} {(y - z)}^{6}} \sqrt[5]{x^{6} {(y - z)}^{12}}$

Step 1

Rewrite $x^{3} {(y - z)}^{6}$ as ${(y + {(- 1)}^{5} z)}^{5} (x^{3} (y - z))$ .

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Step 1.1

Factor out ${(y - z)}^{5}$ .

$\sqrt[5]{x^{3} ({(y - z)}^{5} (y - z))} \sqrt[5]{x^{6} {(y - z)}^{12}}$

Step 1.2

Reorder $x^{3}$ and ${(y - z)}^{5}$ .

$\sqrt[5]{{(y - z)}^{5} x^{3} (y - z)} \sqrt[5]{x^{6} {(y - z)}^{12}}$

Step 1.3

Rewrite $- 1$ as ${(- 1)}^{5}$ .

$\sqrt[5]{{(y + {(- 1)}^{5} z)}^{5} x^{3} (y - z)} \sqrt[5]{x^{6} {(y - z)}^{12}}$

Step 1.4

Add parentheses.

$\sqrt[5]{{(y + {(- 1)}^{5} z)}^{5} (x^{3} (y - z))} \sqrt[5]{x^{6} {(y - z)}^{12}}$

Step 2

Pull terms out from under the radical.

$(y + {(- 1)}^{5} z) \sqrt[5]{x^{3} (y - z)} \sqrt[5]{x^{6} {(y - z)}^{12}}$

Step 3

Rewrite $x^{6} {(y - z)}^{12}$ as ${(x {(y - z)}^{2})}^{5} (x {(y - z)}^{2})$ .

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Step 3.1

Factor out $x^{5}$ .

$(y + {(- 1)}^{5} z) \sqrt[5]{x^{3} (y - z)} \sqrt[5]{x^{5} x {(y - z)}^{12}}$

Step 3.2

Factor out ${(y - z)}^{10}$ .

$(y + {(- 1)}^{5} z) \sqrt[5]{x^{3} (y - z)} \sqrt[5]{x^{5} x ({(y - z)}^{10} {(y - z)}^{2})}$

Step 3.3

Rewrite ${(y - z)}^{10}$ as ${({(y - z)}^{2})}^{5}$ .

$(y + {(- 1)}^{5} z) \sqrt[5]{x^{3} (y - z)} \sqrt[5]{x^{5} x ({({(y - z)}^{2})}^{5} {(y - z)}^{2})}$

Step 3.4

Move $x$ .

$(y + {(- 1)}^{5} z) \sqrt[5]{x^{3} (y - z)} \sqrt[5]{x^{5} {({(y - z)}^{2})}^{5} x {(y - z)}^{2}}$

Step 3.5

Rewrite $x^{5} {({(y - z)}^{2})}^{5}$ as ${(x {(y - z)}^{2})}^{5}$ .

$(y + {(- 1)}^{5} z) \sqrt[5]{x^{3} (y - z)} \sqrt[5]{{(x {(y - z)}^{2})}^{5} x {(y - z)}^{2}}$

Step 3.6

Add parentheses.

$(y + {(- 1)}^{5} z) \sqrt[5]{x^{3} (y - z)} \sqrt[5]{{(x {(y - z)}^{2})}^{5} (x {(y - z)}^{2})}$

Step 4

Pull terms out from under the radical.

$(y + {(- 1)}^{5} z) \sqrt[5]{x^{3} (y - z)} (x {(y - z)}^{2} \sqrt[5]{x {(y - z)}^{2}})$

Step 5

Combine using the product rule for radicals.

$(y + {(- 1)}^{5} z) (x {(y - z)}^{2} \sqrt[5]{x^{3} (y - z) (x {(y - z)}^{2})})$

Step 6

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

$(y + {(- 1)}^{5} z) (x {(y - z)}^{2} \sqrt[5]{x^{1} x^{3} (y - z) {(y - z)}^{2}})$

Step 7

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

$(y + {(- 1)}^{5} z) (x {(y - z)}^{2} \sqrt[5]{x^{1 + 3} (y - z) {(y - z)}^{2}})$

Step 8

Add $1$ and $3$ .

$(y + {(- 1)}^{5} z) (x {(y - z)}^{2} \sqrt[5]{x^{4} (y - z) {(y - z)}^{2}})$

Step 9

Multiply $y - z$ by ${(y - z)}^{2}$ by adding the exponents.

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Step 9.1

Move ${(y - z)}^{2}$ .

$(y + {(- 1)}^{5} z) (x {(y - z)}^{2} \sqrt[5]{x^{4} ({(y - z)}^{2} (y - z))})$

Step 9.2

Multiply ${(y - z)}^{2}$ by $y - z$ .

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Step 9.2.1

Raise $y - z$ to the power of $1$ .

$(y + {(- 1)}^{5} z) (x {(y - z)}^{2} \sqrt[5]{x^{4} ({(y - z)}^{2} {(y - z)}^{1})})$

Step 9.2.2

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

$(y + {(- 1)}^{5} z) (x {(y - z)}^{2} \sqrt[5]{x^{4} {(y - z)}^{2 + 1}})$

Step 9.3

Add $2$ and $1$ .

$(y + {(- 1)}^{5} z) (x {(y - z)}^{2} \sqrt[5]{x^{4} {(y - z)}^{3}})$