Solve frac{frac{xsqrt{x}+8}{x-2sqrt{x}+4}}{frac{x+2sqrt{x}+4}{xsqrt{x}-8}}

$$\frac{ \frac{ x \sqrt{ x } +8 }{ x-2 \sqrt{ x } +4 } }{ \frac{ x+2 \sqrt{ x } +4 }{ x \sqrt{ x } -8 } }$$

Evaluate

$$x-4$$

Short Solution Steps

Divide \frac{x\sqrt{x}+8}{x-2\sqrt{x}+4} by \frac{x+2\sqrt{x}+4}{x\sqrt{x}-8} by multiplying \frac{x\sqrt{x}+8}{x-2\sqrt{x}+4} by the reciprocal of \frac{x+2\sqrt{x}+4}{x\sqrt{x}-8}.

$$\frac{\left(x\sqrt{x}+8\right)\left(x\sqrt{x}-8\right)}{\left(x-2\sqrt{x}+4\right)\left(x+2\sqrt{x}+4\right)}$$

Consider \left(x\sqrt{x}+8\right)\left(x\sqrt{x}-8\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{\left(x\sqrt{x}\right)^{2}-8^{2}}{\left(x-2\sqrt{x}+4\right)\left(x+2\sqrt{x}+4\right)}$$

Expand \left(x\sqrt{x}\right)^{2}.

$$\frac{x^{2}\left(\sqrt{x}\right)^{2}-8^{2}}{\left(x-2\sqrt{x}+4\right)\left(x+2\sqrt{x}+4\right)}$$

Calculate \sqrt{x} to the power of 2 and get x.

$$\frac{x^{2}x-8^{2}}{\left(x-2\sqrt{x}+4\right)\left(x+2\sqrt{x}+4\right)}$$

To multiply powers of the same base, add their exponents. Add 2 and 1 to get 3.

$$\frac{x^{3}-8^{2}}{\left(x-2\sqrt{x}+4\right)\left(x+2\sqrt{x}+4\right)}$$

Calculate 8 to the power of 2 and get 64.

$$\frac{x^{3}-64}{\left(x-2\sqrt{x}+4\right)\left(x+2\sqrt{x}+4\right)}$$

Apply the distributive property by multiplying each term of x-2\sqrt{x}+4 by each term of x+2\sqrt{x}+4.

$$\frac{x^{3}-64}{\left(x-2\sqrt{x}\right)x+2\left(x-2\sqrt{x}\right)\sqrt{x}+4\left(x-2\sqrt{x}\right)+4x+8\sqrt{x}+16}$$

Use the distributive property to multiply x-2\sqrt{x} by x.

$$\frac{x^{3}-64}{x^{2}-2\sqrt{x}x+2\left(x-2\sqrt{x}\right)\sqrt{x}+4\left(x-2\sqrt{x}\right)+4x+8\sqrt{x}+16}$$

Use the distributive property to multiply 2 by x-2\sqrt{x}.

$$\frac{x^{3}-64}{x^{2}-2\sqrt{x}x+\left(2x-4\sqrt{x}\right)\sqrt{x}+4\left(x-2\sqrt{x}\right)+4x+8\sqrt{x}+16}$$

Use the distributive property to multiply 2x-4\sqrt{x} by \sqrt{x}.

$$\frac{x^{3}-64}{x^{2}-2\sqrt{x}x+2x\sqrt{x}-4\left(\sqrt{x}\right)^{2}+4\left(x-2\sqrt{x}\right)+4x+8\sqrt{x}+16}$$

Calculate \sqrt{x} to the power of 2 and get x.

$$\frac{x^{3}-64}{x^{2}-2\sqrt{x}x+2x\sqrt{x}-4x+4\left(x-2\sqrt{x}\right)+4x+8\sqrt{x}+16}$$

Combine -2\sqrt{x}x and 2x\sqrt{x} to get 0.

$$\frac{x^{3}-64}{x^{2}-4x+4\left(x-2\sqrt{x}\right)+4x+8\sqrt{x}+16}$$

Use the distributive property to multiply 4 by x-2\sqrt{x}.

$$\frac{x^{3}-64}{x^{2}-4x+4x-8\sqrt{x}+4x+8\sqrt{x}+16}$$

Combine -4x and 4x to get 0.

$$\frac{x^{3}-64}{x^{2}-8\sqrt{x}+4x+8\sqrt{x}+16}$$

Combine -8\sqrt{x} and 8\sqrt{x} to get 0.

$$\frac{x^{3}-64}{x^{2}+4x+16}$$

Factor the expressions that are not already factored.

$$\frac{\left(x-4\right)\left(x^{2}+4x+16\right)}{x^{2}+4x+16}$$

Cancel out x^{2}+4x+16 in both numerator and denominator.

$$x-4$$

Differentiate w.r.t. x

$$1$$

Steps Using Derivative Rule for Sum

Divide \frac{x\sqrt{x}+8}{x-2\sqrt{x}+4} by \frac{x+2\sqrt{x}+4}{x\sqrt{x}-8} by multiplying \frac{x\sqrt{x}+8}{x-2\sqrt{x}+4} by the reciprocal of \frac{x+2\sqrt{x}+4}{x\sqrt{x}-8}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{\left(x\sqrt{x}+8\right)\left(x\sqrt{x}-8\right)}{\left(x-2\sqrt{x}+4\right)\left(x+2\sqrt{x}+4\right)})$$

Consider \left(x\sqrt{x}+8\right)\left(x\sqrt{x}-8\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{\left(x\sqrt{x}\right)^{2}-8^{2}}{\left(x-2\sqrt{x}+4\right)\left(x+2\sqrt{x}+4\right)})$$

Expand \left(x\sqrt{x}\right)^{2}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{x^{2}\left(\sqrt{x}\right)^{2}-8^{2}}{\left(x-2\sqrt{x}+4\right)\left(x+2\sqrt{x}+4\right)})$$

Calculate \sqrt{x} to the power of 2 and get x.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{x^{2}x-8^{2}}{\left(x-2\sqrt{x}+4\right)\left(x+2\sqrt{x}+4\right)})$$

To multiply powers of the same base, add their exponents. Add 2 and 1 to get 3.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{x^{3}-8^{2}}{\left(x-2\sqrt{x}+4\right)\left(x+2\sqrt{x}+4\right)})$$

Calculate 8 to the power of 2 and get 64.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{x^{3}-64}{\left(x-2\sqrt{x}+4\right)\left(x+2\sqrt{x}+4\right)})$$

Apply the distributive property by multiplying each term of x-2\sqrt{x}+4 by each term of x+2\sqrt{x}+4.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{x^{3}-64}{\left(x-2\sqrt{x}\right)x+2\left(x-2\sqrt{x}\right)\sqrt{x}+4\left(x-2\sqrt{x}\right)+4x+8\sqrt{x}+16})$$

Use the distributive property to multiply x-2\sqrt{x} by x.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{x^{3}-64}{x^{2}-2\sqrt{x}x+2\left(x-2\sqrt{x}\right)\sqrt{x}+4\left(x-2\sqrt{x}\right)+4x+8\sqrt{x}+16})$$

Use the distributive property to multiply 2 by x-2\sqrt{x}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{x^{3}-64}{x^{2}-2\sqrt{x}x+\left(2x-4\sqrt{x}\right)\sqrt{x}+4\left(x-2\sqrt{x}\right)+4x+8\sqrt{x}+16})$$

Use the distributive property to multiply 2x-4\sqrt{x} by \sqrt{x}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{x^{3}-64}{x^{2}-2\sqrt{x}x+2x\sqrt{x}-4\left(\sqrt{x}\right)^{2}+4\left(x-2\sqrt{x}\right)+4x+8\sqrt{x}+16})$$

Calculate \sqrt{x} to the power of 2 and get x.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{x^{3}-64}{x^{2}-2\sqrt{x}x+2x\sqrt{x}-4x+4\left(x-2\sqrt{x}\right)+4x+8\sqrt{x}+16})$$

Combine -2\sqrt{x}x and 2x\sqrt{x} to get 0.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{x^{3}-64}{x^{2}-4x+4\left(x-2\sqrt{x}\right)+4x+8\sqrt{x}+16})$$

Use the distributive property to multiply 4 by x-2\sqrt{x}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{x^{3}-64}{x^{2}-4x+4x-8\sqrt{x}+4x+8\sqrt{x}+16})$$

Combine -4x and 4x to get 0.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{x^{3}-64}{x^{2}-8\sqrt{x}+4x+8\sqrt{x}+16})$$

Combine -8\sqrt{x} and 8\sqrt{x} to get 0.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{x^{3}-64}{x^{2}+4x+16})$$

Factor the expressions that are not already factored in \frac{x^{3}-64}{x^{2}+4x+16}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{\left(x-4\right)\left(x^{2}+4x+16\right)}{x^{2}+4x+16})$$

Cancel out x^{2}+4x+16 in both numerator and denominator.

$$\frac{\mathrm{d}}{\mathrm{d}x}(x-4)$$

The derivative of a polynomial is the sum of the derivatives of its terms. The derivative of a constant term is 0. The derivative of ax^{n} is nax^{n-1}.

$$x^{1-1}$$

Subtract 1 from 1.

$$x^{0}$$

For any term t except 0, t^{0}=1.

$$1$$