Question

simplify the expression completely if possible.
\\(\dfrac{6x}{24x^4 - 168x^3}\\)

Explanation:

Step1: Factor the denominator

First, we factor out the greatest common factor (GCF) from the denominator \(24x^{4}-168x^{3}\). The GCF of \(24x^{4}\) and \(168x^{3}\) is \(24x^{3}\)? Wait, no, let's check again. The GCF of 24 and 168 is 24? Wait, 24 times 7 is 168? No, 24 times 7 is 168? Wait, 247 = 168? 247 = 168, yes. And the GCF of \(x^{4}\) and \(x^{3}\) is \(x^{3}\). So the GCF of \(24x^{4}\) and \(168x^{3}\) is \(24x^{3}\)? Wait, no, 24 and 168: let's find GCF of 24 and 168. Prime factors of 24: \(2^3\times3\), prime factors of 168: \(2^3\times3\times7\). So GCF is \(2^3\times3 = 24\). So GCF of \(24x^{4}\) and \(168x^{3}\) is \(24x^{3}\)? Wait, \(24x^{4}=24x^{3}\times x\), \(168x^{3}=24x^{3}\times7\)? Wait, 247 is 168? 247=168, yes. So \(24x^{4}-168x^{3}=24x^{3}(x - 7)\)? Wait, no, \(24x^{3}\times x=24x^{4}\), \(24x^{3}\times7 = 168x^{3}\), so \(24x^{4}-168x^{3}=24x^{3}(x - 7)\)? Wait, but 24x^{4}-168x^{3}=24x^{3}(x)-168x^{3}=x^{3}(24x - 168)=x^{3}\times24(x - 7)=24x^{3}(x - 7). Wait, but let's check the numerator: 6x. So let's factor denominator correctly. Wait, maybe I made a mistake. Let's factor denominator again. \(24x^{4}-168x^{3}\). Let's factor out 24x^{3}? Wait, 24x^{4} - 168x^{3} = 24x^{3}(x) - 168x^{3} = x^{3}(24x - 168) = x^{3}\times24(x - 7) = 24x^{3}(x - 7). But numerator is 6x. So let's see, 6x and 24x^{3}(x - 7). Let's factor numerator and denominator.

Numerator: \(6x\)

Denominator: \(24x^{4}-168x^{3}=24x^{3}(x - 7)\)? Wait, no, 24x^{4}-168x^{3}=24x^{3}(x) - 168x^{3}=x^{3}(24x - 168)=x^{3}\times24(x - 7)=24x^{3}(x - 7). Wait, but 24x - 168 = 24(x - 7), yes. So denominator is \(24x^{3}(x - 7)\).

Wait, but let's check GCF of 6x and 24x^{3}(x - 7). The GCF of 6 and 24 is 6, and GCF of x and x^{3} is x. So GCF is 6x.

So let's factor numerator and denominator:

Numerator: \(6x\)

Denominator: \(24x^{4}-168x^{3}=24x^{3}(x - 7)=6x\times4x^{2}(x - 7)\)

So now, we can cancel out the common factors.

Step2: Simplify the fraction

\(\frac{6x}{24x^{4}-168x^{3}}=\frac{6x}{24x^{3}(x - 7)}\) (Wait, no, earlier I thought denominator is 24x^{3}(x - 7), but let's check 24x^{3}(x - 7)=24x^{4}-168x^{3}, yes. So numerator is 6x, denominator is 24x^{3}(x - 7). Now, we can simplify the coefficients and the variables.

Coefficients: 6 and 24. 6/24 = 1/4.

Variables: x in numerator and x^{3} in denominator. x/x^{3} = 1/x^{2}.

So putting it together: \(\frac{6x}{24x^{3}(x - 7)}=\frac{6}{24}\times\frac{x}{x^{3}}\times\frac{1}{x - 7}=\frac{1}{4}\times\frac{1}{x^{2}}\times\frac{1}{x - 7}=\frac{1}{4x^{2}(x - 7)}\)

Wait, but let's check again. Let's factor denominator correctly. Wait, 24x^{4}-168x^{3}=24x^{3}(x) - 168x^{3}=x^{3}(24x - 168)=x^{3}\times24(x - 7)=24x^{3}(x - 7). Numerator is 6x. So GCF of 6x and 24x^{3}(x - 7) is 6x. So divide numerator and denominator by 6x:

Numerator: 6x ÷ 6x = 1

Denominator: 24x^{3}(x - 7) ÷ 6x = 4x^{2}(x - 7)

So the simplified expression is \(\frac{1}{4x^{2}(x - 7)}\)

Wait, but let's verify. Let's expand the denominator: 4x^{2}(x - 7)=4x^{3}-28x^{2}. Wait, no, original denominator is 24x^{4}-168x^{3}=24x^{3}(x - 7). So when we divide numerator and denominator by 6x, we get (6x)/(24x^{3}(x - 7)) = 1/(4x^{2}(x - 7)). Yes, that's correct.

Answer:

\(\boxed{\dfrac{1}{4x^2(x - 7)}}\)