Solve (ax^n)(bx^m)=(ab)x^n+m

$$( a x ^ { n } ) ( b x ^ { m } ) = ( a b ) x ^ { n + m }$$

Solve for a (complex solution)

$$a\in \mathrm{C}$$

Steps for Solving Linear Equation

Subtract abx^{n+m} from both sides.

$$ax^{n}bx^{m}-abx^{n+m}=0$$

Reorder the terms.

$$abx^{m}x^{n}-abx^{m+n}=0$$

Combine all terms containing a.

$$\left(bx^{m}x^{n}-bx^{m+n}\right)a=0$$

The equation is in standard form.

$$\text{true}$$

This is true for any a.

$$a\in \mathrm{R}$$

Solve for b (complex solution)

$$b\in \mathrm{C}$$

Steps for Solving Linear Equation

Subtract abx^{n+m} from both sides.

$$ax^{n}bx^{m}-abx^{n+m}=0$$

Reorder the terms.

$$abx^{m}x^{n}-abx^{m+n}=0$$

Combine all terms containing b.

$$\left(ax^{m}x^{n}-ax^{m+n}\right)b=0$$

The equation is in standard form.

$$\text{true}$$

This is true for any b.

$$b\in \mathrm{R}$$

Solve for a

$$a\in \mathrm{R}$$

$$\left(x<0\text{ and }Denominator(n)\text{bmod}2=1\text{ and }Denominator(m)\text{bmod}2=1\text{ and }Denominator(m+n)\text{bmod}2=1\right)\text{ or }\left(x=0\text{ and }n>0\text{ and }m>0\right)\text{ or }x>0$$

Steps for Solving Linear Equation

Subtract abx^{n+m} from both sides.

$$ax^{n}bx^{m}-abx^{n+m}=0$$

Reorder the terms.

$$abx^{m}x^{n}-abx^{m+n}=0$$

Combine all terms containing a.

$$\left(bx^{m}x^{n}-bx^{m+n}\right)a=0$$

The equation is in standard form.

$$\text{true}$$

This is true for any a.

$$a\in \mathrm{R}$$

Solve for b

$$b\in \mathrm{R}$$

$$\left(x<0\text{ and }Denominator(n)\text{bmod}2=1\text{ and }Denominator(m)\text{bmod}2=1\text{ and }Denominator(m+n)\text{bmod}2=1\right)\text{ or }\left(x=0\text{ and }n>0\text{ and }m>0\right)\text{ or }x>0$$

Steps for Solving Linear Equation

Subtract abx^{n+m} from both sides.

$$ax^{n}bx^{m}-abx^{n+m}=0$$

Reorder the terms.

$$abx^{m}x^{n}-abx^{m+n}=0$$

Combine all terms containing b.

$$\left(ax^{m}x^{n}-ax^{m+n}\right)b=0$$

The equation is in standard form.

$$\text{true}$$

This is true for any b.

$$b\in \mathrm{R}$$