Simplify -4/(((15x^2)/8)/(25x^3))
The problem is asking you to perform operations on a complex fraction to simplify it. The fraction involves negative numbers, multiplication, division, and exponentiation of a variable (x). Simplifying complex fractions generally includes finding a common denominator, performing division on fractions, simplifying exponents, and reducing the fractions to their simplest form. The goal is to rewrite the expression so that it is easier to understand or work with by eliminating the complex fraction structure.
$\frac{- 4}{\frac{\frac{15 x^{2}}{8}}{25 x^{3}}}$
Answer
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: $-4 \times \left(25x^3 \times \frac{8}{15x^2}\right)$
Step 3
Identify and simplify common factors:
Step 3.1
Extract $5x^2$ from $25x^3$: $-4 \times \left(5x^2(5x) \times \frac{8}{15x^2}\right)$
Step 3.2
Extract $5x^2$ from $15x^2$: $-4 \times \left(5x^2(5x) \times \frac{8}{5x^2(3)}\right)$
Step 3.3
Eliminate the common $5x^2$ terms: $-4 \times \left(\cancel{5x^2}(5x) \times \frac{8}{\cancel{5x^2}(3)}\right)$
Step 3.4
Rewrite the simplified expression: $-4 \times \left(5x \times \frac{8}{3}\right)$
Step 4
Combine the constants 5 and $\frac{8}{3}$: $-4 \times \left(x \times \frac{5 \cdot 8}{3}\right)$
Step 5
Perform the multiplication of 5 and 8: $-4 \times \left(x \times \frac{40}{3}\right)$
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:
Multiplication by Reciprocal: When dividing by a fraction, you can multiply by its reciprocal (i.e., flip the numerator and denominator).
Simplifying Common Factors: If there are common factors in the numerator and denominator, they can be canceled out to simplify the expression.
Combining Constants: Constants can be combined through multiplication or division as appropriate.
Multiplying Terms: When multiplying terms, you can rearrange and group them to simplify the calculation process.
Negative Signs: Negative signs can be moved around in a fraction as long as the overall value of the expression remains unchanged.
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.
