Solve (5+4sqrt{2})^2+(8sqrt{2}-5)^2+15+100)

$$( 5 + 4 \sqrt { 2 } ) ^ { 2 } + ( 8 \sqrt { 2 } - 5 ) ^ { 2 } + 15 + 100 )$$

Evaluate

$$325-40\sqrt{2}\approx 268.431457505$$

Solution Steps

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

$$25+40\sqrt{2}+16\left(\sqrt{2}\right)^{2}+\left(8\sqrt{2}-5\right)^{2}+15+100$$

The square of \sqrt{2} is 2.

$$25+40\sqrt{2}+16\times 2+\left(8\sqrt{2}-5\right)^{2}+15+100$$

Multiply 16 and 2 to get 32.

$$25+40\sqrt{2}+32+\left(8\sqrt{2}-5\right)^{2}+15+100$$

Add 25 and 32 to get 57.

$$57+40\sqrt{2}+\left(8\sqrt{2}-5\right)^{2}+15+100$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(8\sqrt{2}-5\right)^{2}.

$$57+40\sqrt{2}+64\left(\sqrt{2}\right)^{2}-80\sqrt{2}+25+15+100$$

The square of \sqrt{2} is 2.

$$57+40\sqrt{2}+64\times 2-80\sqrt{2}+25+15+100$$

Multiply 64 and 2 to get 128.

$$57+40\sqrt{2}+128-80\sqrt{2}+25+15+100$$

Add 128 and 25 to get 153.

$$57+40\sqrt{2}+153-80\sqrt{2}+15+100$$

Add 57 and 153 to get 210.

$$210+40\sqrt{2}-80\sqrt{2}+15+100$$

Combine 40\sqrt{2} and -80\sqrt{2} to get -40\sqrt{2}.

$$210-40\sqrt{2}+15+100$$

Add 210 and 15 to get 225.

$$225-40\sqrt{2}+100$$

Add 225 and 100 to get 325.

$$325-40\sqrt{2}$$

Expand

$$325-40\sqrt{2}$$

Solution Steps

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

$$25+40\sqrt{2}+16\left(\sqrt{2}\right)^{2}+\left(8\sqrt{2}-5\right)^{2}+15+100$$

The square of \sqrt{2} is 2.

$$25+40\sqrt{2}+16\times 2+\left(8\sqrt{2}-5\right)^{2}+15+100$$

Multiply 16 and 2 to get 32.

$$25+40\sqrt{2}+32+\left(8\sqrt{2}-5\right)^{2}+15+100$$

Add 25 and 32 to get 57.

$$57+40\sqrt{2}+\left(8\sqrt{2}-5\right)^{2}+15+100$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(8\sqrt{2}-5\right)^{2}.

$$57+40\sqrt{2}+64\left(\sqrt{2}\right)^{2}-80\sqrt{2}+25+15+100$$

The square of \sqrt{2} is 2.

$$57+40\sqrt{2}+64\times 2-80\sqrt{2}+25+15+100$$

Multiply 64 and 2 to get 128.

$$57+40\sqrt{2}+128-80\sqrt{2}+25+15+100$$

Add 128 and 25 to get 153.

$$57+40\sqrt{2}+153-80\sqrt{2}+15+100$$

Add 57 and 153 to get 210.

$$210+40\sqrt{2}-80\sqrt{2}+15+100$$

Combine 40\sqrt{2} and -80\sqrt{2} to get -40\sqrt{2}.

$$210-40\sqrt{2}+15+100$$

Add 210 and 15 to get 225.

$$225-40\sqrt{2}+100$$

Add 225 and 100 to get 325.

$$325-40\sqrt{2}$$