Solve {left(-32xy/25+16y^2right)}^2-4*(16x^2-400)/25+16y^2

$${ \left( \frac{ -32xy }{ 25+16 { y }^{ 2 } } \right) }^{ 2 } - \frac{ 4 \times (16 { x }^{ 2 } -400) }{ 25+16 { y }^{ 2 } }$$

Evaluate

$$\frac{1600\left(25+16y^{2}-x^{2}\right)}{\left(16y^{2}+25\right)^{2}}$$

Short Solution Steps

To raise \frac{-32xy}{25+16y^{2}} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{\left(-32xy\right)^{2}}{\left(25+16y^{2}\right)^{2}}-\frac{4\left(16x^{2}-400\right)}{25+16y^{2}}$$

Use the distributive property to multiply 4 by 16x^{2}-400.

$$\frac{\left(-32xy\right)^{2}}{\left(25+16y^{2}\right)^{2}}-\frac{64x^{2}-1600}{25+16y^{2}}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(25+16y^{2}\right)^{2} and 25+16y^{2} is \left(16y^{2}+25\right)^{2}. Multiply \frac{64x^{2}-1600}{25+16y^{2}} times \frac{16y^{2}+25}{16y^{2}+25}.

$$\frac{\left(-32xy\right)^{2}}{\left(16y^{2}+25\right)^{2}}-\frac{\left(64x^{2}-1600\right)\left(16y^{2}+25\right)}{\left(16y^{2}+25\right)^{2}}$$

Since \frac{\left(-32xy\right)^{2}}{\left(16y^{2}+25\right)^{2}} and \frac{\left(64x^{2}-1600\right)\left(16y^{2}+25\right)}{\left(16y^{2}+25\right)^{2}} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\left(-32xy\right)^{2}-\left(64x^{2}-1600\right)\left(16y^{2}+25\right)}{\left(16y^{2}+25\right)^{2}}$$

Do the multiplications in \left(-32xy\right)^{2}-\left(64x^{2}-1600\right)\left(16y^{2}+25\right).

$$\frac{\left(-32xy\right)^{2}-1024x^{2}y^{2}-1600x^{2}+25600y^{2}+40000}{\left(16y^{2}+25\right)^{2}}$$

Combine like terms in \left(-32xy\right)^{2}-1024x^{2}y^{2}-1600x^{2}+25600y^{2}+40000.

$$\frac{25600y^{2}-1600x^{2}+40000}{\left(16y^{2}+25\right)^{2}}$$

Expand \left(16y^{2}+25\right)^{2}.

$$\frac{25600y^{2}-1600x^{2}+40000}{256y^{4}+800y^{2}+625}$$

Expand

$$\frac{1600\left(25+16y^{2}-x^{2}\right)}{\left(16y^{2}+25\right)^{2}}$$

Short Solution Steps

To raise \frac{-32xy}{25+16y^{2}} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{\left(-32xy\right)^{2}}{\left(25+16y^{2}\right)^{2}}-\frac{4\left(16x^{2}-400\right)}{25+16y^{2}}$$

Use the distributive property to multiply 4 by 16x^{2}-400.

$$\frac{\left(-32xy\right)^{2}}{\left(25+16y^{2}\right)^{2}}-\frac{64x^{2}-1600}{25+16y^{2}}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(25+16y^{2}\right)^{2} and 25+16y^{2} is \left(16y^{2}+25\right)^{2}. Multiply \frac{64x^{2}-1600}{25+16y^{2}} times \frac{16y^{2}+25}{16y^{2}+25}.

$$\frac{\left(-32xy\right)^{2}}{\left(16y^{2}+25\right)^{2}}-\frac{\left(64x^{2}-1600\right)\left(16y^{2}+25\right)}{\left(16y^{2}+25\right)^{2}}$$

Since \frac{\left(-32xy\right)^{2}}{\left(16y^{2}+25\right)^{2}} and \frac{\left(64x^{2}-1600\right)\left(16y^{2}+25\right)}{\left(16y^{2}+25\right)^{2}} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\left(-32xy\right)^{2}-\left(64x^{2}-1600\right)\left(16y^{2}+25\right)}{\left(16y^{2}+25\right)^{2}}$$

Do the multiplications in \left(-32xy\right)^{2}-\left(64x^{2}-1600\right)\left(16y^{2}+25\right).

$$\frac{\left(-32xy\right)^{2}-1024x^{2}y^{2}-1600x^{2}+25600y^{2}+40000}{\left(16y^{2}+25\right)^{2}}$$

Combine like terms in \left(-32xy\right)^{2}-1024x^{2}y^{2}-1600x^{2}+25600y^{2}+40000.

$$\frac{25600y^{2}-1600x^{2}+40000}{\left(16y^{2}+25\right)^{2}}$$

Expand \left(16y^{2}+25\right)^{2}.

$$\frac{25600y^{2}-1600x^{2}+40000}{256y^{4}+800y^{2}+625}$$