Simplify 1/(4x^2)-(2x-4)/(16x^3)
The question is asking to perform the operation of simplification on a given algebraic expression. The expression consists of two fractions that involve variables, specifically "x" with exponents. The simplification process would likely involve finding a common denominator in order to combine the two fractions into one and then simplifying the result further, if possible, by canceling out common factors in the numerator and the denominator or by simplifying the expression within the numerator itself.
$\frac{1}{4 x^{2}} - \frac{2 x - 4}{16 x^{3}}$
Answer
Solution:
Step:1
Eliminate the common factors between the numerator and the denominator.
Step:1.1
Extract the factor of $2$ from $2x$.
$$\frac{1}{4x^2} - \frac{2(x) - 4}{16x^3}$$
Step:1.2
Extract the factor of $2$ from $-4$.
$$\frac{1}{4x^2} - \frac{2x + 2 \cdot (-2)}{16x^3}$$
Step:1.3
Extract the factor of $2$ from $2x + 2 \cdot (-2)$.
$$\frac{1}{4x^2} - \frac{2(x - 2)}{16x^3}$$
Step:1.4
Eliminate the common factors.
Step:1.4.1
Extract the factor of $2$ from $16x^3$.
$$\frac{1}{4x^2} - \frac{2(x - 2)}{2(8x^3)}$$
Step:1.4.2
Eliminate the common factor.
$$\frac{1}{4x^2} - \frac{\cancel{2}(x - 2)}{\cancel{2}(8x^3)}$$
Step:1.4.3
Reformulate the expression.
$$\frac{1}{4x^2} - \frac{x - 2}{8x^3}$$
Step:2
To express $\frac{1}{4x^2}$ with a common denominator, multiply by $\frac{2x}{2x}$.
$$\frac{1}{4x^2} \cdot \frac{2x}{2x} - \frac{x - 2}{8x^3}$$
Step:3
Represent each term with a common denominator of $8x^3$, by multiplying each by an appropriate form of $1$.
Step:3.1
Multiply $\frac{1}{4x^2}$ by $\frac{2x}{2x}$.
$$\frac{2x}{4x^2(2x)} - \frac{x - 2}{8x^3}$$
Step:3.2
Reorder using the commutative property of multiplication.
$$\frac{2x}{4 \cdot 2x^2x} - \frac{x - 2}{8x^3}$$
Step:3.3
Combine $x^2$ and $x$ by adding their exponents.
Step:3.3.1
Rearrange $x$.
$$\frac{2x}{4 \cdot 2(x \cdot x^2)} - \frac{x - 2}{8x^3}$$
Step:3.3.2
Multiply $x$ by $x^2$.
Step:3.3.2.1
Express $x$ as $x^1$.
$$\frac{2x}{4 \cdot 2(x^1 \cdot x^2)} - \frac{x - 2}{8x^3}$$
Step:3.3.2.2
Apply the exponent rule $a^m \cdot a^n = a^{m+n}$ to combine the exponents.
$$\frac{2x}{4 \cdot 2x^{1+2}} - \frac{x - 2}{8x^3}$$
Step:3.3.3
Add the exponents $1$ and $2$.
$$\frac{2x}{4 \cdot 2x^3} - \frac{x - 2}{8x^3}$$
Step:3.4
Multiply $4$ by $2$.
$$\frac{2x}{8x^3} - \frac{x - 2}{8x^3}$$
Step:4
Consolidate the numerators over the shared denominator.
$$\frac{2x - (x - 2)}{8x^3}$$
Step:5
Simplify the numerator.
Step:5.1
Apply the distributive property.
$$\frac{2x - x - (-2)}{8x^3}$$
Step:5.2
Multiply $-1$ by $-2$.
$$\frac{2x - x + 2}{8x^3}$$
Step:5.3
Subtract $x$ from $2x$.
$$\frac{x + 2}{8x^3}$$
Knowledge Notes:
To simplify the given expression $\frac{1}{4x^2} - \frac{2x - 4}{16x^3}$, we follow these steps:
Factorization: Identify and factor out common factors in the numerators and denominators to simplify the expression.
Common Denominator: Find a common denominator for the terms in the expression to combine them into a single fraction.
Distributive Property: Use the distributive property to expand or simplify expressions, particularly when dealing with subtraction or addition of polynomials.
Exponent Rules: Apply exponent rules such as $a^m \cdot a^n = a^{m+n}$ to simplify expressions involving variables raised to powers.
Combining Like Terms: Combine like terms, which are terms that have the same variable raised to the same power, to simplify the expression further.
By following these steps and applying the properties of arithmetic and algebra, we can simplify complex rational expressions into a more manageable form.
