Question

Simplify the fraction. $$\frac{56 x^{2} y^{5}}{64 x^{2} y}$$

Short answer

The simplified form of given fraction is \( \frac{7 y^{4}}{8} \).

Step-by-step solution

Identifying Common factors

In the given expression \( \frac{56 x^{2} y^{5}}{64 x^{2} y} \), one can observe that both numerator and denominator share 2 common variables \( x^{2} \) and \( y \). Also, we can simplify the coefficient of the fraction (56 and 64) by dividing them by their greatest common divisor (8).

Cancel out Similar Variables

By the law of exponents, \( x^{a} / x^{b} = x^{(a-b)} \) and \( y^{a} / y^{b} = y^{(a-b)} \). When we apply this rule, \( x^{2} / x^{2} = x^{(2-2)} = x^{0} \) and \( y^{5} / y = y^{(5-1)} = y^{4} \). Since any non-zero number to the power of zero is 1, \( x^{0} = 1 \). Thus, the simplification results in \( 1 \cdot \frac{56 y^{4}}{64} = \frac{56 y^{4}}{64} \)

Simplify the Coefficient of the Fraction

We continue to simplify the fraction by reducing the coefficient. Given that 56 and 64 are divisible by 8. Dividing them, we get \( \frac{7 y^{4}}{8} \).