Simplify ((x^3)/(5x^15))/((5x)/(25x^4))

Solution:

Step 1

Eliminate the shared $x^3$ term from both $x^3$ and $x^{15}$.

Step 1.1

Introduce a multiplicative identity of 1. $\frac{\frac{x^{3} \cdot 1}{5 x^{15}}}{\frac{5 x}{25 x^{4}}}$

Step 1.2

Remove common factors.

Step 1.2.1

Extract $x^3$ from $5 x^{15}$. $\frac{\frac{x^{3} \cdot 1}{x^{3}(5 x^{12})}}{\frac{5 x}{25 x^{4}}}$

Step 1.2.2

Eliminate the shared $x^3$ term. $\frac{\frac{\cancel{x^{3}} \cdot 1}{\cancel{x^{3}}(5 x^{12})}}{\frac{5 x}{25 x^{4}}}$

Step 1.2.3

Reformulate the expression. $\frac{\frac{1}{5 x^{12}}}{\frac{5 x}{25 x^{4}}}$

Step 2

Remove the common $5$ from $5$ and $25$.

Step 2.1

Isolate $5$ from $5 x$. $\frac{\frac{1}{5 x^{12}}}{\frac{5(x)}{25 x^{4}}}$

Step 2.2

Eliminate common factors.

Step 2.2.1

Extract $5$ from $25 x^{4}$. $\frac{\frac{1}{5 x^{12}}}{\frac{5(x)}{5(5 x^{4})}}$

Step 2.2.2

Remove the shared $5$. $\frac{\frac{1}{5 x^{12}}}{\frac{\cancel{5} x}{\cancel{5}(5 x^{4})}}$

Step 2.2.3

Reformulate the expression. $\frac{\frac{1}{5 x^{12}}}{\frac{x}{5 x^{4}}}$

Step 3

Eliminate the shared $x$ from $x$ and $x^4$.

Step 3.1

Raise $x$ to the first power. $\frac{\frac{1}{5 x^{12}}}{\frac{x^1}{5 x^{4}}}$

Step 3.2

Isolate $x$ from $x^1$. $\frac{\frac{1}{5 x^{12}}}{\frac{x \cdot 1}{5 x^{4}}}$

Step 3.3

Remove common factors.

Step 3.3.1

Extract $x$ from $5 x^{4}$. $\frac{\frac{1}{5 x^{12}}}{\frac{x \cdot 1}{x(5 x^{3})}}$

Step 3.3.2

Eliminate the shared $x$. $\frac{\frac{1}{5 x^{12}}}{\frac{\cancel{x} \cdot 1}{\cancel{x}(5 x^{3})}}$

Step 3.3.3

Reformulate the expression. $\frac{\frac{1}{5 x^{12}}}{\frac{1}{5 x^{3}}}$

Step 4

Multiply the numerator by the reciprocal of the denominator. $\frac{1}{5 x^{12}}(5 x^{3})$

Step 5

Apply the commutative property of multiplication. $5 \frac{1}{5 x^{12}} x^{3}$

Step 6

Eliminate the common $5$.

Step 6.1

Isolate $5$ from $5 x^{12}$. $5 \frac{1}{5(x^{12})} x^{3}$

Step 6.2

Remove the shared $5$. $\cancel{5} \frac{1}{\cancel{5} x^{12}} x^{3}$

Step 6.3

Reformulate the expression. $\frac{1}{x^{12}} x^{3}$

Step 7

Cancel the common $x^3$ term.

Step 7.1

Extract $x^3$ from $x^{12}$. $\frac{1}{x^3 x^9} x^3$

Step 7.2

Eliminate the shared $x^3$. $\frac{1}{\cancel{x^3} x^9} \cancel{x^3}$

Step 7.3

Reformulate the expression. $\frac{1}{x^9}$

Knowledge Notes:

The problem involves simplifying a complex fraction, which is a fraction where the numerator or the denominator (or both) is also a fraction. The steps to simplify such expressions typically involve:

1. Factorization: Breaking down expressions into their constituent factors to identify and cancel out common terms.

2. Multiplication by the reciprocal: To divide by a fraction, you multiply by its reciprocal (i.e., you flip the numerator and the denominator).

3. Exponent rules: When dividing terms with the same base, you subtract the exponents (e.g., $x^m / x^n = x^{m-n}$).

4. Commutative property of multiplication: This property states that the order in which two numbers are multiplied does not affect the product (e.g., $ab = ba$).

5. Cancellation: If a term appears in both the numerator and the denominator, it can be cancelled out, reducing the expression to its simplest form.

In the given problem, these principles are applied in a step-by-step manner to simplify the complex fraction to $\frac{1}{x^9}$.