Solve f(x)=(-3x^{-3}+3x^{5})^3(x^2-x^3)^2

$$f ( x ) = ( - 3 x ^ { - 3 } + 3 x ^ { 5 } ) ^ { 3 } ( x ^ { 2 } - x ^ { 3 } ) ^ { 2 }$$

Evaluate

$$\frac{27\left(1-x\right)^{2}\left(x^{8}-1\right)^{3}}{x^{5}}$$

Solution Steps

Use binomial theorem \left(a+b\right)^{3}=a^{3}+3a^{2}b+3ab^{2}+b^{3} to expand \left(-3x^{-3}+3x^{5}\right)^{3}.

$$\left(-27\left(x^{-3}\right)^{3}+81\left(x^{-3}\right)^{2}x^{5}-81x^{-3}\left(x^{5}\right)^{2}+27\left(x^{5}\right)^{3}\right)\left(x^{2}-x^{3}\right)^{2}$$

To raise a power to another power, multiply the exponents. Multiply -3 and 3 to get -9.

$$\left(-27x^{-9}+81\left(x^{-3}\right)^{2}x^{5}-81x^{-3}\left(x^{5}\right)^{2}+27\left(x^{5}\right)^{3}\right)\left(x^{2}-x^{3}\right)^{2}$$

To raise a power to another power, multiply the exponents. Multiply -3 and 2 to get -6.

$$\left(-27x^{-9}+81x^{-6}x^{5}-81x^{-3}\left(x^{5}\right)^{2}+27\left(x^{5}\right)^{3}\right)\left(x^{2}-x^{3}\right)^{2}$$

To multiply powers of the same base, add their exponents. Add -6 and 5 to get -1.

$$\left(-27x^{-9}+81x^{-1}-81x^{-3}\left(x^{5}\right)^{2}+27\left(x^{5}\right)^{3}\right)\left(x^{2}-x^{3}\right)^{2}$$

To raise a power to another power, multiply the exponents. Multiply 5 and 2 to get 10.

$$\left(-27x^{-9}+81x^{-1}-81x^{-3}x^{10}+27\left(x^{5}\right)^{3}\right)\left(x^{2}-x^{3}\right)^{2}$$

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

$$\left(-27x^{-9}+81x^{-1}-81x^{7}+27\left(x^{5}\right)^{3}\right)\left(x^{2}-x^{3}\right)^{2}$$

To raise a power to another power, multiply the exponents. Multiply 5 and 3 to get 15.

$$\left(-27x^{-9}+81x^{-1}-81x^{7}+27x^{15}\right)\left(x^{2}-x^{3}\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x^{2}-x^{3}\right)^{2}.

$$\left(-27x^{-9}+81x^{-1}-81x^{7}+27x^{15}\right)\left(\left(x^{2}\right)^{2}-2x^{2}x^{3}+\left(x^{3}\right)^{2}\right)$$

To raise a power to another power, multiply the exponents. Multiply 2 and 2 to get 4.

$$\left(-27x^{-9}+81x^{-1}-81x^{7}+27x^{15}\right)\left(x^{4}-2x^{2}x^{3}+\left(x^{3}\right)^{2}\right)$$

To multiply powers of the same base, add their exponents. Add 2 and 3 to get 5.

$$\left(-27x^{-9}+81x^{-1}-81x^{7}+27x^{15}\right)\left(x^{4}-2x^{5}+\left(x^{3}\right)^{2}\right)$$

To raise a power to another power, multiply the exponents. Multiply 3 and 2 to get 6.

$$\left(-27x^{-9}+81x^{-1}-81x^{7}+27x^{15}\right)\left(x^{4}-2x^{5}+x^{6}\right)$$

Use the distributive property to multiply -27x^{-9}+81x^{-1}-81x^{7}+27x^{15} by x^{4}-2x^{5}+x^{6}.

$$-27x^{-9}x^{4}+54x^{-4}-27x^{-9}x^{6}+81x^{-1}x^{4}-162x^{4}+81x^{-1}x^{6}-81x^{11}+162x^{12}-81x^{13}+27x^{19}-54x^{20}+27x^{21}$$

To multiply powers of the same base, add their exponents. Add -9 and 4 to get -5.

$$-27x^{-5}+54x^{-4}-27x^{-9}x^{6}+81x^{-1}x^{4}-162x^{4}+81x^{-1}x^{6}-81x^{11}+162x^{12}-81x^{13}+27x^{19}-54x^{20}+27x^{21}$$

To multiply powers of the same base, add their exponents. Add -9 and 6 to get -3.

$$-27x^{-5}+54x^{-4}-27x^{-3}+81x^{-1}x^{4}-162x^{4}+81x^{-1}x^{6}-81x^{11}+162x^{12}-81x^{13}+27x^{19}-54x^{20}+27x^{21}$$

To multiply powers of the same base, add their exponents. Add -1 and 4 to get 3.

$$-27x^{-5}+54x^{-4}-27x^{-3}+81x^{3}-162x^{4}+81x^{-1}x^{6}-81x^{11}+162x^{12}-81x^{13}+27x^{19}-54x^{20}+27x^{21}$$

To multiply powers of the same base, add their exponents. Add -1 and 6 to get 5.

$$-27x^{-5}+54x^{-4}-27x^{-3}+81x^{3}-162x^{4}+81x^{5}-81x^{11}+162x^{12}-81x^{13}+27x^{19}-54x^{20}+27x^{21}$$

Expand

$$\frac{27\left(x^{26}-2x^{25}+x^{24}-3x^{18}+6x^{17}-3x^{16}+3x^{10}-6x^{9}+3x^{8}-x^{2}+2x-1\right)}{x^{5}}$$

Solution Steps

Use binomial theorem \left(a+b\right)^{3}=a^{3}+3a^{2}b+3ab^{2}+b^{3} to expand \left(-3x^{-3}+3x^{5}\right)^{3}.

$$\left(-27\left(x^{-3}\right)^{3}+81\left(x^{-3}\right)^{2}x^{5}-81x^{-3}\left(x^{5}\right)^{2}+27\left(x^{5}\right)^{3}\right)\left(x^{2}-x^{3}\right)^{2}$$

To raise a power to another power, multiply the exponents. Multiply -3 and 3 to get -9.

$$\left(-27x^{-9}+81\left(x^{-3}\right)^{2}x^{5}-81x^{-3}\left(x^{5}\right)^{2}+27\left(x^{5}\right)^{3}\right)\left(x^{2}-x^{3}\right)^{2}$$

To raise a power to another power, multiply the exponents. Multiply -3 and 2 to get -6.

$$\left(-27x^{-9}+81x^{-6}x^{5}-81x^{-3}\left(x^{5}\right)^{2}+27\left(x^{5}\right)^{3}\right)\left(x^{2}-x^{3}\right)^{2}$$

To multiply powers of the same base, add their exponents. Add -6 and 5 to get -1.

$$\left(-27x^{-9}+81x^{-1}-81x^{-3}\left(x^{5}\right)^{2}+27\left(x^{5}\right)^{3}\right)\left(x^{2}-x^{3}\right)^{2}$$

To raise a power to another power, multiply the exponents. Multiply 5 and 2 to get 10.

$$\left(-27x^{-9}+81x^{-1}-81x^{-3}x^{10}+27\left(x^{5}\right)^{3}\right)\left(x^{2}-x^{3}\right)^{2}$$

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

$$\left(-27x^{-9}+81x^{-1}-81x^{7}+27\left(x^{5}\right)^{3}\right)\left(x^{2}-x^{3}\right)^{2}$$

To raise a power to another power, multiply the exponents. Multiply 5 and 3 to get 15.

$$\left(-27x^{-9}+81x^{-1}-81x^{7}+27x^{15}\right)\left(x^{2}-x^{3}\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x^{2}-x^{3}\right)^{2}.

$$\left(-27x^{-9}+81x^{-1}-81x^{7}+27x^{15}\right)\left(\left(x^{2}\right)^{2}-2x^{2}x^{3}+\left(x^{3}\right)^{2}\right)$$

To raise a power to another power, multiply the exponents. Multiply 2 and 2 to get 4.

$$\left(-27x^{-9}+81x^{-1}-81x^{7}+27x^{15}\right)\left(x^{4}-2x^{2}x^{3}+\left(x^{3}\right)^{2}\right)$$

To multiply powers of the same base, add their exponents. Add 2 and 3 to get 5.

$$\left(-27x^{-9}+81x^{-1}-81x^{7}+27x^{15}\right)\left(x^{4}-2x^{5}+\left(x^{3}\right)^{2}\right)$$

To raise a power to another power, multiply the exponents. Multiply 3 and 2 to get 6.

$$\left(-27x^{-9}+81x^{-1}-81x^{7}+27x^{15}\right)\left(x^{4}-2x^{5}+x^{6}\right)$$

Use the distributive property to multiply -27x^{-9}+81x^{-1}-81x^{7}+27x^{15} by x^{4}-2x^{5}+x^{6}.

$$-27x^{-9}x^{4}+54x^{-4}-27x^{-9}x^{6}+81x^{-1}x^{4}-162x^{4}+81x^{-1}x^{6}-81x^{11}+162x^{12}-81x^{13}+27x^{19}-54x^{20}+27x^{21}$$

To multiply powers of the same base, add their exponents. Add -9 and 4 to get -5.

$$-27x^{-5}+54x^{-4}-27x^{-9}x^{6}+81x^{-1}x^{4}-162x^{4}+81x^{-1}x^{6}-81x^{11}+162x^{12}-81x^{13}+27x^{19}-54x^{20}+27x^{21}$$

To multiply powers of the same base, add their exponents. Add -9 and 6 to get -3.

$$-27x^{-5}+54x^{-4}-27x^{-3}+81x^{-1}x^{4}-162x^{4}+81x^{-1}x^{6}-81x^{11}+162x^{12}-81x^{13}+27x^{19}-54x^{20}+27x^{21}$$

To multiply powers of the same base, add their exponents. Add -1 and 4 to get 3.

$$-27x^{-5}+54x^{-4}-27x^{-3}+81x^{3}-162x^{4}+81x^{-1}x^{6}-81x^{11}+162x^{12}-81x^{13}+27x^{19}-54x^{20}+27x^{21}$$

To multiply powers of the same base, add their exponents. Add -1 and 6 to get 5.

$$-27x^{-5}+54x^{-4}-27x^{-3}+81x^{3}-162x^{4}+81x^{5}-81x^{11}+162x^{12}-81x^{13}+27x^{19}-54x^{20}+27x^{21}$$