Simplify (4x^3*2x^4y^10)/(50x^9y^4)

Solution:

Step:1 Combine the powers of x by summing their exponents.

Step:1.1 Combine $x^3$ and $x^4$ in the numerator.$\frac{4(x^3{x}^4)\cdot (2y^{10})}{50x^9{y}^4}$

Step:1.2 Apply the exponent rule $a^m\cdot a^n=a^{m+n}$.$\frac{4x^{3+4}\cdot (2y^{10})}{50x^9{y}^4}$

Step:1.3 Calculate the sum of the exponents.$\frac{4x^7\cdot (2y^{10})}{50x^9{y}^4}$

Step:2 Simplify by reducing the common factors in the numerator and denominator.

Step:2.1 Extract the factor of 2 from the numerator.$\frac{2(2x^7\cdot (2y^{10}))}{50x^9{y}^4}$

Step:2.2 Reduce the common factors.

Step:2.2.1 Extract the factor of 2 from the denominator.$\frac{2(2x^7\cdot (2y^{10}))}{2(25x^9{y}^4)}$

Step:2.2.2 Eliminate the common factor of 2.$\frac{\cancel{2}(2x^7\cdot (2y^{10}))}{\cancel{2}(25x^9{y}^4)}$

Step:2.2.3 Present the simplified expression.$\frac{2x^7\cdot (2y^{10})}{25x^9{y}^4}$

Step:3 Reduce the common x terms by subtracting exponents.

Step:3.1 Extract $x^7$ from the numerator.$\frac{x^7(2\cdot (2y^{10}))}{25x^9{y}^4}$

Step:3.2 Reduce the common x terms.

Step:3.2.1 Extract $x^7$ from the denominator.$\frac{x^7(2\cdot (2y^{10}))}{x^7(25x^2{y}^4)}$

Step:3.2.2 Eliminate the common factor of $x^7$.$\frac{\cancel{x^7}(2\cdot (2y^{10}))}{\cancel{x^7}(25x^2{y}^4)}$

Step:3.2.3 Present the simplified expression.$\frac{2\cdot (2y^{10})}{25x^2{y}^4}$

Step:4 Reduce the common y terms by subtracting exponents.

Step:4.1 Extract $y^4$ from the numerator.$\frac{y^4(2\cdot (2y^6))}{25x^2{y}^4}$

Step:4.2 Reduce the common y terms.

Step:4.2.1 Extract $y^4$ from the denominator.$\frac{y^4(2\cdot (2y^6))}{y^4(25x^2)}$

Step:4.2.2 Eliminate the common factor of $y^4$.$\frac{\cancel{y^4}(2\cdot (2y^6))}{\cancel{y^4}(25x^2)}$

Step:4.2.3 Present the simplified expression.$\frac{2\cdot (2y^6)}{25x^2}$

Step:5 Final simplification of the expression.

Step:5.1 Multiply the constants in the numerator.$\frac{2^2{y}^6}{25x^2}$

Step:5.2 Calculate the power of 2.$\frac{4y^6}{25x^2}$

Knowledge Notes:

The problem involves simplifying a rational expression with polynomial terms in both the numerator and the denominator. The steps taken to simplify the expression are based on the following algebraic rules and properties:

1. Exponent Rules: When multiplying like bases, you add the exponents ($a^m\cdot a^n=a^{m+n}$). When dividing like bases, you subtract the exponents ($\frac{a^m}{a^n}=a^{m-n}$ when $m\ge n$).

2. Common Factor Reduction: If a factor appears in both the numerator and the denominator, it can be cancelled out. This is because any number divided by itself equals 1.

3. Multiplication of Constants: Constants can be multiplied together and can also be factored out if they appear as a common factor in both the numerator and the denominator.

4. Final Simplification: After all common factors have been cancelled and the exponents have been simplified, the final step is to present the expression in its simplest form.

These rules are applied systematically in each step to simplify the given algebraic fraction to its simplest form.