Solve x^5(2x^2-3x^-5)+x^-3(1/{x^{-10}}+frac{5{x}^{-2}}{{x}^-5})+frac{3{x}^5}{frac{{x}^3}{{x}^-2}}

$${ x }^{ 5 } (2 { x }^{ 2 } -3 { x }^{ -5 } )+ { x }^{ -3 } ( \frac{ 1 }{ { x }^{ -10 } } + \frac{ 5 { x }^{ -2 } }{ { x }^{ -5 } } )+ \frac{ 3 { x }^{ 5 } }{ \frac{ { x }^{ 3 } }{ { x }^{ -2 } } }$$

Evaluate

$$3x^{7}+5$$

Short Solution Steps

To divide powers of the same base, subtract the denominator's exponent from the numerator's exponent. Subtract -2 from 3 to get 5.

$$x^{5}\left(2x^{2}-3x^{-5}\right)+x^{-3}\left(\frac{1}{x^{-10}}+\frac{5x^{-2}}{x^{-5}}\right)+\frac{3x^{5}}{x^{5}}$$

To divide powers of the same base, subtract the denominator's exponent from the numerator's exponent.

$$x^{5}\left(2x^{2}-3x^{-5}\right)+x^{-3}\left(\frac{1}{x^{-10}}+5x^{3}\right)+\frac{3x^{5}}{x^{5}}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 5x^{3} times \frac{x^{-10}}{x^{-10}}.

$$x^{5}\left(2x^{2}-3x^{-5}\right)+x^{-3}\left(\frac{1}{x^{-10}}+\frac{5x^{3}x^{-10}}{x^{-10}}\right)+\frac{3x^{5}}{x^{5}}$$

Since \frac{1}{x^{-10}} and \frac{5x^{3}x^{-10}}{x^{-10}} have the same denominator, add them by adding their numerators.

$$x^{5}\left(2x^{2}-3x^{-5}\right)+x^{-3}\times \frac{1+5x^{3}x^{-10}}{x^{-10}}+\frac{3x^{5}}{x^{5}}$$

Do the multiplications in 1+5x^{3}x^{-10}.

$$x^{5}\left(2x^{2}-3x^{-5}\right)+x^{-3}\times \frac{1+5x^{-7}}{x^{-10}}+\frac{3x^{5}}{x^{5}}$$

Factor the expressions that are not already factored in \frac{1+5x^{-7}}{x^{-10}}.

$$x^{5}\left(2x^{2}-3x^{-5}\right)+x^{-3}\times \frac{x^{-7}\left(x^{7}+5\right)}{x^{-10}}+\frac{3x^{5}}{x^{5}}$$

To divide powers of the same base, subtract the denominator's exponent from the numerator's exponent.

$$x^{5}\left(2x^{2}-3x^{-5}\right)+x^{-3}x^{3}\left(x^{7}+5\right)+\frac{3x^{5}}{x^{5}}$$

Expand the expression.

$$x^{5}\left(2x^{2}-3x^{-5}\right)+x^{-3}\left(x^{10}+5x^{3}\right)+\frac{3x^{5}}{x^{5}}$$

Cancel out x^{5} in both numerator and denominator.

$$x^{5}\left(2x^{2}-3x^{-5}\right)+x^{-3}\left(x^{10}+5x^{3}\right)+3$$

Use the distributive property to multiply x^{5} by 2x^{2}-3x^{-5}.

$$2x^{7}-3x^{5}x^{-5}+x^{-3}\left(x^{10}+5x^{3}\right)+3$$

Multiply x^{5} and x^{-5} to get 1.

$$2x^{7}-3+x^{-3}\left(x^{10}+5x^{3}\right)+3$$

Use the distributive property to multiply x^{-3} by x^{10}+5x^{3}.

$$2x^{7}-3+x^{-3}x^{10}+5x^{-3}x^{3}+3$$

To multiply powers of the same base, add their exponents. Add -3 and 10 to get 7.

$$2x^{7}-3+x^{7}+5x^{-3}x^{3}+3$$

Multiply x^{-3} and x^{3} to get 1.

$$2x^{7}-3+x^{7}+5+3$$

Combine 2x^{7} and x^{7} to get 3x^{7}.

$$3x^{7}-3+5+3$$

Add -3 and 5 to get 2.

$$3x^{7}+2+3$$

Add 2 and 3 to get 5.

$$3x^{7}+5$$

Expand

$$3x^{7}+5$$

Short Solution Steps

To divide powers of the same base, subtract the denominator's exponent from the numerator's exponent. Subtract -2 from 3 to get 5.

$$x^{5}\left(2x^{2}-3x^{-5}\right)+x^{-3}\left(\frac{1}{x^{-10}}+\frac{5x^{-2}}{x^{-5}}\right)+\frac{3x^{5}}{x^{5}}$$

To divide powers of the same base, subtract the denominator's exponent from the numerator's exponent.

$$x^{5}\left(2x^{2}-3x^{-5}\right)+x^{-3}\left(\frac{1}{x^{-10}}+5x^{3}\right)+\frac{3x^{5}}{x^{5}}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 5x^{3} times \frac{x^{-10}}{x^{-10}}.

$$x^{5}\left(2x^{2}-3x^{-5}\right)+x^{-3}\left(\frac{1}{x^{-10}}+\frac{5x^{3}x^{-10}}{x^{-10}}\right)+\frac{3x^{5}}{x^{5}}$$

Since \frac{1}{x^{-10}} and \frac{5x^{3}x^{-10}}{x^{-10}} have the same denominator, add them by adding their numerators.

$$x^{5}\left(2x^{2}-3x^{-5}\right)+x^{-3}\times \frac{1+5x^{3}x^{-10}}{x^{-10}}+\frac{3x^{5}}{x^{5}}$$

Do the multiplications in 1+5x^{3}x^{-10}.

$$x^{5}\left(2x^{2}-3x^{-5}\right)+x^{-3}\times \frac{1+5x^{-7}}{x^{-10}}+\frac{3x^{5}}{x^{5}}$$

Factor the expressions that are not already factored in \frac{1+5x^{-7}}{x^{-10}}.

$$x^{5}\left(2x^{2}-3x^{-5}\right)+x^{-3}\times \frac{x^{-7}\left(x^{7}+5\right)}{x^{-10}}+\frac{3x^{5}}{x^{5}}$$

To divide powers of the same base, subtract the denominator's exponent from the numerator's exponent.

$$x^{5}\left(2x^{2}-3x^{-5}\right)+x^{-3}x^{3}\left(x^{7}+5\right)+\frac{3x^{5}}{x^{5}}$$

Expand the expression.

$$x^{5}\left(2x^{2}-3x^{-5}\right)+x^{-3}\left(x^{10}+5x^{3}\right)+\frac{3x^{5}}{x^{5}}$$

Cancel out x^{5} in both numerator and denominator.

$$x^{5}\left(2x^{2}-3x^{-5}\right)+x^{-3}\left(x^{10}+5x^{3}\right)+3$$

Use the distributive property to multiply x^{5} by 2x^{2}-3x^{-5}.

$$2x^{7}-3x^{5}x^{-5}+x^{-3}\left(x^{10}+5x^{3}\right)+3$$

Multiply x^{5} and x^{-5} to get 1.

$$2x^{7}-3+x^{-3}\left(x^{10}+5x^{3}\right)+3$$

Use the distributive property to multiply x^{-3} by x^{10}+5x^{3}.

$$2x^{7}-3+x^{-3}x^{10}+5x^{-3}x^{3}+3$$

To multiply powers of the same base, add their exponents. Add -3 and 10 to get 7.

$$2x^{7}-3+x^{7}+5x^{-3}x^{3}+3$$

Multiply x^{-3} and x^{3} to get 1.

$$2x^{7}-3+x^{7}+5+3$$

Combine 2x^{7} and x^{7} to get 3x^{7}.

$$3x^{7}-3+5+3$$

Add -3 and 5 to get 2.

$$3x^{7}+2+3$$

Add 2 and 3 to get 5.

$$3x^{7}+5$$