Solve in Terms of the Arbitrary Variable n (2x^9y^n)(4x^2y^10)=8x^11y^20

$(2 x^{9} y^{n}) (4 x^{2} y^{10}) = 8 x^{11} y^{20}$

Step 1

Solve the equation for $x$ .

Simplify $2 x^{9} y^{n} (4 x^{2} y^{10})$ .

Multiply $x^{9}$ by $x^{2}$ by adding the exponents.

Move $x^{2}$ .

$2 (x^{2} x^{9}) y^{n} (4 y^{10}) = 8 x^{11} y^{20}$

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

$2 x^{2 + 9} y^{n} (4 y^{10}) = 8 x^{11} y^{20}$

Add $2$ and $9$ .

$2 x^{11} y^{n} (4 y^{10}) = 8 x^{11} y^{20}$

Multiply $y^{n}$ by $y^{10}$ by adding the exponents.

Move $y^{10}$ .

$2 x^{11} (y^{10} y^{n}) \cdot 4 = 8 x^{11} y^{20}$

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

$2 x^{11} y^{10 + n} \cdot 4 = 8 x^{11} y^{20}$

Multiply $4$ by $2$ .

$8 x^{11} y^{10 + n} = 8 x^{11} y^{20}$

Subtract $8 x^{11} y^{20}$ from both sides of the equation.

$8 x^{11} y^{10 + n} - 8 x^{11} y^{20} = 0$

Factor $8 x^{11}$ out of $8 x^{11} y^{10 + n} - 8 x^{11} y^{20}$ .

Factor $8 x^{11}$ out of $8 x^{11} y^{10 + n}$ .

$8 x^{11} (y^{10 + n}) - 8 x^{11} y^{20} = 0$

Factor $8 x^{11}$ out of $- 8 x^{11} y^{20}$ .

$8 x^{11} (y^{10 + n}) + 8 x^{11} (- 1 y^{20}) = 0$

Factor $8 x^{11}$ out of $8 x^{11} (y^{10 + n}) + 8 x^{11} (- 1 y^{20})$ .

$8 x^{11} (y^{10 + n} - 1 y^{20}) = 0$

Rewrite $- 1 y^{20}$ as $- y^{20}$ .

$8 x^{11} (y^{10 + n} - y^{20}) = 0$

Divide each term in $8 x^{11} (y^{10 + n} - y^{20}) = 0$ by $8 (y^{10 + n} - y^{20})$ and simplify.

Divide each term in $8 x^{11} (y^{10 + n} - y^{20}) = 0$ by $8 (y^{10 + n} - y^{20})$ .

$\frac{8 x^{11} (y^{10 + n} - y^{20})}{8 (y^{10 + n} - y^{20})} = \frac{0}{8 (y^{10 + n} - y^{20})}$

Simplify the left side.

Cancel the common factor of $8$ .

Cancel the common factor.

$\frac{8 x^{11} (y^{10 + n} - y^{20})}{8 (y^{10 + n} - y^{20})} = \frac{0}{8 (y^{10 + n} - y^{20})}$

Rewrite the expression.

$\frac{x^{11} (y^{10 + n} - y^{20})}{y^{10 + n} - y^{20}} = \frac{0}{8 (y^{10 + n} - y^{20})}$

Cancel the common factor of $y^{10 + n} - y^{20}$ .

Cancel the common factor.

$\frac{x^{11} (y^{10 + n} - y^{20})}{y^{10 + n} - y^{20}} = \frac{0}{8 (y^{10 + n} - y^{20})}$

Divide $x^{11}$ by $1$ .

$x^{11} = \frac{0}{8 (y^{10 + n} - y^{20})}$

Simplify the right side.

Cancel the common factor of $0$ and $8$ .

Factor $8$ out of $0$ .

$x^{11} = \frac{8 (0)}{8 (y^{10 + n} - y^{20})}$

Cancel the common factors.

Cancel the common factor.

$x^{11} = \frac{8 \cdot 0}{8 (y^{10 + n} - y^{20})}$

Rewrite the expression.

$x^{11} = \frac{0}{y^{10 + n} - y^{20}}$

Divide $0$ by $y^{10 + n} - y^{20}$ .

$x^{11} = 0$

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \sqrt[11]{0}$

Simplify $\sqrt[11]{0}$ .

Rewrite $0$ as $0^{11}$ .

$x = \sqrt[11]{0^{11}}$

Pull terms out from under the radical, assuming real numbers.

$x = 0$

Step 2

Solve the equation for $y$ .

Simplify $2 \cdot (0^{9} y^{n} (4 \cdot (0^{2} y^{10})))$ .

Multiply $0^{9}$ by $0^{2}$ by adding the exponents.

Move $0^{2}$ .

$2 \cdot (0^{2} \cdot 0^{9} y^{n} (4 \cdot (y^{10}))) = 8 \cdot (0^{11} y^{20})$

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

$2 \cdot (0^{2 + 9} y^{n} (4 \cdot (y^{10}))) = 8 \cdot (0^{11} y^{20})$

Add $2$ and $9$ .

$2 \cdot (0^{11} y^{n} (4 \cdot (y^{10}))) = 8 \cdot (0^{11} y^{20})$

Multiply $y^{n}$ by $y^{10}$ by adding the exponents.

Move $y^{10}$ .

$2 \cdot (0^{11} (y^{10} y^{n}) \cdot 4) = 8 \cdot (0^{11} y^{20})$

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

$2 \cdot (0^{11} y^{10 + n} \cdot 4) = 8 \cdot (0^{11} y^{20})$

Raising $0$ to any positive power yields $0$ .

$2 \cdot (0 y^{10 + n} \cdot 4) = 8 \cdot (0^{11} y^{20})$

Multiply $0$ by $y^{10 + n}$ .

$2 \cdot (0 \cdot 4) = 8 \cdot (0^{11} y^{20})$

Multiply $2 (0 \cdot 4)$ .

Multiply $0$ by $4$ .

$2 \cdot 0 = 8 \cdot (0^{11} y^{20})$

Multiply $2$ by $0$ .

$0 = 8 \cdot (0^{11} y^{20})$

Simplify $8 \cdot (0^{11} y^{20})$ .

Raising $0$ to any positive power yields $0$ .

$0 = 8 \cdot (0 y^{20})$

Multiply $0$ by $y^{20}$ .

$0 = 8 \cdot 0$

Multiply $8$ by $0$ .

$0 = 0$

Since $0 = 0$ , the equation will always be true.

Always true

Solve in Terms of the Arbitrary Variable n (2x^9y^n)(4x^2y^10)=8x^11y^20

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