Solve y=x^4(3x^7+4x^5)^3

$$y = x ^ { 4 } ( 3 x ^ { 7 } + 4 x ^ { 5 } ) ^ { 3 }$$

Solve for y

$$y=\left(3x^{2}+4\right)^{3}x^{19}$$

Solution Steps

Use binomial theorem \left(a+b\right)^{3}=a^{3}+3a^{2}b+3ab^{2}+b^{3} to expand \left(3x^{7}+4x^{5}\right)^{3}.

$$y=x^{4}\left(27\left(x^{7}\right)^{3}+108\left(x^{7}\right)^{2}x^{5}+144x^{7}\left(x^{5}\right)^{2}+64\left(x^{5}\right)^{3}\right)$$

To raise a power to another power, multiply the exponents. Multiply 7 and 3 to get 21.

$$y=x^{4}\left(27x^{21}+108\left(x^{7}\right)^{2}x^{5}+144x^{7}\left(x^{5}\right)^{2}+64\left(x^{5}\right)^{3}\right)$$

To raise a power to another power, multiply the exponents. Multiply 7 and 2 to get 14.

$$y=x^{4}\left(27x^{21}+108x^{14}x^{5}+144x^{7}\left(x^{5}\right)^{2}+64\left(x^{5}\right)^{3}\right)$$

To multiply powers of the same base, add their exponents. Add 14 and 5 to get 19.

$$y=x^{4}\left(27x^{21}+108x^{19}+144x^{7}\left(x^{5}\right)^{2}+64\left(x^{5}\right)^{3}\right)$$

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

$$y=x^{4}\left(27x^{21}+108x^{19}+144x^{7}x^{10}+64\left(x^{5}\right)^{3}\right)$$

To multiply powers of the same base, add their exponents. Add 7 and 10 to get 17.

$$y=x^{4}\left(27x^{21}+108x^{19}+144x^{17}+64\left(x^{5}\right)^{3}\right)$$

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

$$y=x^{4}\left(27x^{21}+108x^{19}+144x^{17}+64x^{15}\right)$$

Use the distributive property to multiply x^{4} by 27x^{21}+108x^{19}+144x^{17}+64x^{15}.

$$y=27x^{25}+108x^{23}+144x^{21}+64x^{19}$$