Solve (227x/136y+89/56)(227/136yx-89/56)

$$( \frac { 227 x } { 136 y } + \frac { 89 } { 56 } ) ( \frac { 227 } { 136 y } x - \frac { 89 } { 56 } )$$

Evaluate

$$\frac{51529\times \left(\frac{x}{y}\right)^{2}}{18496}-\frac{7921}{3136}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 136y and 56 is 952y. Multiply \frac{227x}{136y} times \frac{7}{7}. Multiply \frac{89}{56} times \frac{17y}{17y}.

$$\left(\frac{7\times 227x}{952y}+\frac{89\times 17y}{952y}\right)\left(\frac{227}{136y}x-\frac{89}{56}\right)$$

Since \frac{7\times 227x}{952y} and \frac{89\times 17y}{952y} have the same denominator, add them by adding their numerators.

$$\frac{7\times 227x+89\times 17y}{952y}\left(\frac{227}{136y}x-\frac{89}{56}\right)$$

Do the multiplications in 7\times 227x+89\times 17y.

$$\frac{1589x+1513y}{952y}\left(\frac{227}{136y}x-\frac{89}{56}\right)$$

Express \frac{227}{136y}x as a single fraction.

$$\frac{1589x+1513y}{952y}\left(\frac{227x}{136y}-\frac{89}{56}\right)$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 136y and 56 is 952y. Multiply \frac{227x}{136y} times \frac{7}{7}. Multiply \frac{89}{56} times \frac{17y}{17y}.

$$\frac{1589x+1513y}{952y}\left(\frac{7\times 227x}{952y}-\frac{89\times 17y}{952y}\right)$$

Since \frac{7\times 227x}{952y} and \frac{89\times 17y}{952y} have the same denominator, subtract them by subtracting their numerators.

$$\frac{1589x+1513y}{952y}\times \frac{7\times 227x-89\times 17y}{952y}$$

Do the multiplications in 7\times 227x-89\times 17y.

$$\frac{1589x+1513y}{952y}\times \frac{1589x-1513y}{952y}$$

Multiply \frac{1589x+1513y}{952y} times \frac{1589x-1513y}{952y} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\left(1589x+1513y\right)\left(1589x-1513y\right)}{952y\times 952y}$$

Multiply y and y to get y^{2}.

$$\frac{\left(1589x+1513y\right)\left(1589x-1513y\right)}{952y^{2}\times 952}$$

Multiply 952 and 952 to get 906304.

$$\frac{\left(1589x+1513y\right)\left(1589x-1513y\right)}{906304y^{2}}$$

Consider \left(1589x+1513y\right)\left(1589x-1513y\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{\left(1589x\right)^{2}-\left(1513y\right)^{2}}{906304y^{2}}$$

Expand \left(1589x\right)^{2}.

$$\frac{1589^{2}x^{2}-\left(1513y\right)^{2}}{906304y^{2}}$$

Calculate 1589 to the power of 2 and get 2524921.

$$\frac{2524921x^{2}-\left(1513y\right)^{2}}{906304y^{2}}$$

Expand \left(1513y\right)^{2}.

$$\frac{2524921x^{2}-1513^{2}y^{2}}{906304y^{2}}$$

Calculate 1513 to the power of 2 and get 2289169.

$$\frac{2524921x^{2}-2289169y^{2}}{906304y^{2}}$$

Expand

$$\frac{51529\times \left(\frac{x}{y}\right)^{2}}{18496}-\frac{7921}{3136}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 136y and 56 is 952y. Multiply \frac{227x}{136y} times \frac{7}{7}. Multiply \frac{89}{56} times \frac{17y}{17y}.

$$\left(\frac{7\times 227x}{952y}+\frac{89\times 17y}{952y}\right)\left(\frac{227}{136y}x-\frac{89}{56}\right)$$

Since \frac{7\times 227x}{952y} and \frac{89\times 17y}{952y} have the same denominator, add them by adding their numerators.

$$\frac{7\times 227x+89\times 17y}{952y}\left(\frac{227}{136y}x-\frac{89}{56}\right)$$

Do the multiplications in 7\times 227x+89\times 17y.

$$\frac{1589x+1513y}{952y}\left(\frac{227}{136y}x-\frac{89}{56}\right)$$

Express \frac{227}{136y}x as a single fraction.

$$\frac{1589x+1513y}{952y}\left(\frac{227x}{136y}-\frac{89}{56}\right)$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 136y and 56 is 952y. Multiply \frac{227x}{136y} times \frac{7}{7}. Multiply \frac{89}{56} times \frac{17y}{17y}.

$$\frac{1589x+1513y}{952y}\left(\frac{7\times 227x}{952y}-\frac{89\times 17y}{952y}\right)$$

Since \frac{7\times 227x}{952y} and \frac{89\times 17y}{952y} have the same denominator, subtract them by subtracting their numerators.

$$\frac{1589x+1513y}{952y}\times \frac{7\times 227x-89\times 17y}{952y}$$

Do the multiplications in 7\times 227x-89\times 17y.

$$\frac{1589x+1513y}{952y}\times \frac{1589x-1513y}{952y}$$

Multiply \frac{1589x+1513y}{952y} times \frac{1589x-1513y}{952y} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\left(1589x+1513y\right)\left(1589x-1513y\right)}{952y\times 952y}$$

Multiply y and y to get y^{2}.

$$\frac{\left(1589x+1513y\right)\left(1589x-1513y\right)}{952y^{2}\times 952}$$

Multiply 952 and 952 to get 906304.

$$\frac{\left(1589x+1513y\right)\left(1589x-1513y\right)}{906304y^{2}}$$

Consider \left(1589x+1513y\right)\left(1589x-1513y\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{\left(1589x\right)^{2}-\left(1513y\right)^{2}}{906304y^{2}}$$

Expand \left(1589x\right)^{2}.

$$\frac{1589^{2}x^{2}-\left(1513y\right)^{2}}{906304y^{2}}$$

Calculate 1589 to the power of 2 and get 2524921.

$$\frac{2524921x^{2}-\left(1513y\right)^{2}}{906304y^{2}}$$

Expand \left(1513y\right)^{2}.

$$\frac{2524921x^{2}-1513^{2}y^{2}}{906304y^{2}}$$

Calculate 1513 to the power of 2 and get 2289169.

$$\frac{2524921x^{2}-2289169y^{2}}{906304y^{2}}$$