Simplify -4/(((15x^2)/8)/(25x^3))

Solution:

Step 1

Invert the inner fraction and multiply it with the outer numerator: $-4\times \frac{25x^3}{\left(\frac{15x^2}{8}\right)}$

Step 2

Combine the multiplication: $-4\times (25x^3\times \frac{8}{15x^2})$

Step 3

Identify and simplify common factors:

Step 3.1

Extract $5x^2$ from $25x^3$: $-4\times (5x^2(5x)\times \frac{8}{15x^2})$

Step 3.2

Extract $5x^2$ from $15x^2$: $-4\times (5x^2(5x)\times \frac{8}{5x^2(3)})$

Step 3.3

Eliminate the common $5x^2$ terms: $-4\times (\cancel{5x^2}(5x)\times \frac{8}{\cancel{5x^2}(3)})$

Step 3.4

Rewrite the simplified expression: $-4\times (5x\times \frac{8}{3})$

Step 4

Combine the constants 5 and $\frac{8}{3}$: $-4\times (x\times \frac{5\cdot 8}{3})$

Step 5

Perform the multiplication of 5 and 8: $-4\times (x\times \frac{40}{3})$

Step 6

Concatenate $x$ and $\frac{40}{3}$: $-4\times \frac{x\cdot 40}{3}$

Step 7

Rearrange to place 40 before $x$: $-4\times \frac{40\cdot x}{3}$

Step 8

Execute the multiplication of $-4$ and $\frac{40x}{3}$:

Step 8.1

Combine $-4$ with $\frac{40x}{3}$: $\frac{-4(40x)}{3}$

Step 8.2

Multiply $-4$ by 40: $\frac{-160x}{3}$

Step 9

Position the negative sign in front of the fraction: $-\frac{160x}{3}$

Knowledge Notes:

To simplify a complex fraction, you can follow these steps:

1. Multiplication by Reciprocal: When dividing by a fraction, you can multiply by its reciprocal (i.e., flip the numerator and denominator).

2. Simplifying Common Factors: If there are common factors in the numerator and denominator, they can be canceled out to simplify the expression.

3. Combining Constants: Constants can be combined through multiplication or division as appropriate.

4. Multiplying Terms: When multiplying terms, you can rearrange and group them to simplify the calculation process.

5. Negative Signs: Negative signs can be moved around in a fraction as long as the overall value of the expression remains unchanged.

6. LaTeX Formatting: When writing mathematical expressions, LaTeX provides a clear and professional way to present equations and fractions.

In this problem, we used these principles to simplify the given complex fraction step by step, ultimately arriving at a simplified expression.