Solve =(4a-1)^2(4a+1)^2(16a+1)^2

$$= ( 4 a - 1 ) ^ { 2 } ( 4 a + 1 ) ^ { 2 } ( 16 a + 1 ) ^ { 2 }$$

Evaluate

$$\left(\left(16a+1\right)\left(16a^{2}-1\right)\right)^{2}$$

Solution Steps

Use binomial theorem \left(p-q\right)^{2}=p^{2}-2pq+q^{2} to expand \left(4a-1\right)^{2}.

$$\left(16a^{2}-8a+1\right)\left(4a+1\right)^{2}\left(16a+1\right)^{2}$$

Use binomial theorem \left(p+q\right)^{2}=p^{2}+2pq+q^{2} to expand \left(4a+1\right)^{2}.

$$\left(16a^{2}-8a+1\right)\left(16a^{2}+8a+1\right)\left(16a+1\right)^{2}$$

Use binomial theorem \left(p+q\right)^{2}=p^{2}+2pq+q^{2} to expand \left(16a+1\right)^{2}.

$$\left(16a^{2}-8a+1\right)\left(16a^{2}+8a+1\right)\left(256a^{2}+32a+1\right)$$

Use the distributive property to multiply 16a^{2}-8a+1 by 16a^{2}+8a+1 and combine like terms.

$$\left(256a^{4}-32a^{2}+1\right)\left(256a^{2}+32a+1\right)$$

Use the distributive property to multiply 256a^{4}-32a^{2}+1 by 256a^{2}+32a+1 and combine like terms.

$$65536a^{6}+8192a^{5}-7936a^{4}-1024a^{3}+224a^{2}+32a+1$$

Expand

$$65536a^{6}+8192a^{5}-7936a^{4}-1024a^{3}+224a^{2}+32a+1$$

Solution Steps

Use binomial theorem \left(p-q\right)^{2}=p^{2}-2pq+q^{2} to expand \left(4a-1\right)^{2}.

$$\left(16a^{2}-8a+1\right)\left(4a+1\right)^{2}\left(16a+1\right)^{2}$$

Use binomial theorem \left(p+q\right)^{2}=p^{2}+2pq+q^{2} to expand \left(4a+1\right)^{2}.

$$\left(16a^{2}-8a+1\right)\left(16a^{2}+8a+1\right)\left(16a+1\right)^{2}$$

Use binomial theorem \left(p+q\right)^{2}=p^{2}+2pq+q^{2} to expand \left(16a+1\right)^{2}.

$$\left(16a^{2}-8a+1\right)\left(16a^{2}+8a+1\right)\left(256a^{2}+32a+1\right)$$

Use the distributive property to multiply 16a^{2}-8a+1 by 16a^{2}+8a+1 and combine like terms.

$$\left(256a^{4}-32a^{2}+1\right)\left(256a^{2}+32a+1\right)$$

Use the distributive property to multiply 256a^{4}-32a^{2}+1 by 256a^{2}+32a+1 and combine like terms.

$$65536a^{6}+8192a^{5}-7936a^{4}-1024a^{3}+224a^{2}+32a+1$$