Solve [7/8x^5-4/7x^6]^3

$$[ \frac { 7 } { 8 } x ^ { 5 } - \frac { 4 } { 7 } x ^ { 6 } ] ^ { 3 }$$

Evaluate

$$\frac{\left(49-32x\right)^{3}x^{15}}{175616}$$

Solution Steps

Use binomial theorem \left(a-b\right)^{3}=a^{3}-3a^{2}b+3ab^{2}-b^{3} to expand \left(\frac{7}{8}x^{5}-\frac{4}{7}x^{6}\right)^{3}.

$$\frac{343}{512}\left(x^{5}\right)^{3}-\frac{21}{16}\left(x^{5}\right)^{2}x^{6}+\frac{6}{7}x^{5}\left(x^{6}\right)^{2}-\frac{64}{343}\left(x^{6}\right)^{3}$$

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

$$\frac{343}{512}x^{15}-\frac{21}{16}\left(x^{5}\right)^{2}x^{6}+\frac{6}{7}x^{5}\left(x^{6}\right)^{2}-\frac{64}{343}\left(x^{6}\right)^{3}$$

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

$$\frac{343}{512}x^{15}-\frac{21}{16}x^{10}x^{6}+\frac{6}{7}x^{5}\left(x^{6}\right)^{2}-\frac{64}{343}\left(x^{6}\right)^{3}$$

To multiply powers of the same base, add their exponents. Add 10 and 6 to get 16.

$$\frac{343}{512}x^{15}-\frac{21}{16}x^{16}+\frac{6}{7}x^{5}\left(x^{6}\right)^{2}-\frac{64}{343}\left(x^{6}\right)^{3}$$

To raise a power to another power, multiply the exponents. Multiply 6 and 2 to get 12.

$$\frac{343}{512}x^{15}-\frac{21}{16}x^{16}+\frac{6}{7}x^{5}x^{12}-\frac{64}{343}\left(x^{6}\right)^{3}$$

To multiply powers of the same base, add their exponents. Add 5 and 12 to get 17.

$$\frac{343}{512}x^{15}-\frac{21}{16}x^{16}+\frac{6}{7}x^{17}-\frac{64}{343}\left(x^{6}\right)^{3}$$

To raise a power to another power, multiply the exponents. Multiply 6 and 3 to get 18.

$$\frac{343}{512}x^{15}-\frac{21}{16}x^{16}+\frac{6}{7}x^{17}-\frac{64}{343}x^{18}$$

Expand

$$-\frac{64x^{18}}{343}+\frac{6x^{17}}{7}-\frac{21x^{16}}{16}+\frac{343x^{15}}{512}$$

Solution Steps

Use binomial theorem \left(a-b\right)^{3}=a^{3}-3a^{2}b+3ab^{2}-b^{3} to expand \left(\frac{7}{8}x^{5}-\frac{4}{7}x^{6}\right)^{3}.

$$\frac{343}{512}\left(x^{5}\right)^{3}-\frac{21}{16}\left(x^{5}\right)^{2}x^{6}+\frac{6}{7}x^{5}\left(x^{6}\right)^{2}-\frac{64}{343}\left(x^{6}\right)^{3}$$

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

$$\frac{343}{512}x^{15}-\frac{21}{16}\left(x^{5}\right)^{2}x^{6}+\frac{6}{7}x^{5}\left(x^{6}\right)^{2}-\frac{64}{343}\left(x^{6}\right)^{3}$$

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

$$\frac{343}{512}x^{15}-\frac{21}{16}x^{10}x^{6}+\frac{6}{7}x^{5}\left(x^{6}\right)^{2}-\frac{64}{343}\left(x^{6}\right)^{3}$$

To multiply powers of the same base, add their exponents. Add 10 and 6 to get 16.

$$\frac{343}{512}x^{15}-\frac{21}{16}x^{16}+\frac{6}{7}x^{5}\left(x^{6}\right)^{2}-\frac{64}{343}\left(x^{6}\right)^{3}$$

To raise a power to another power, multiply the exponents. Multiply 6 and 2 to get 12.

$$\frac{343}{512}x^{15}-\frac{21}{16}x^{16}+\frac{6}{7}x^{5}x^{12}-\frac{64}{343}\left(x^{6}\right)^{3}$$

To multiply powers of the same base, add their exponents. Add 5 and 12 to get 17.

$$\frac{343}{512}x^{15}-\frac{21}{16}x^{16}+\frac{6}{7}x^{17}-\frac{64}{343}\left(x^{6}\right)^{3}$$

To raise a power to another power, multiply the exponents. Multiply 6 and 3 to get 18.

$$\frac{343}{512}x^{15}-\frac{21}{16}x^{16}+\frac{6}{7}x^{17}-\frac{64}{343}x^{18}$$