Question

simplify the expression completely if possible.
\\(\dfrac{6x}{24x^4 - 168x^3}\\)
answer
attempt 1 out of 2
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Explanation:

Step1: Factor denominator

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: 24 factors are 2×2×2×3, 168 is 2×2×2×3×7. So GCF of 24 and 168 is 24? Wait, no, 24×7=168? No, 24×7=168? 24×7=168? Yes, 24×7=168. Wait, no, 24×7 is 168? 24×7: 20×7=140, 4×7=28, 140+28=168. So GCF of 24 and 168 is 24? Wait, no, 24 and 168: 24 divides into 168 how many times? 168 ÷24=7. So GCF of 24 and 168 is 24? Wait, no, 24 is 2^3×3, 168 is 2^3×3×7. So GCF is 2^3×3=24. And for the variables, \(x^4\) and \(x^3\), GCF is \(x^3\). So the denominator \(24x^4 - 168x^3 = 24x^3(x - 7)\)? Wait, no, 24x^3 times x is 24x^4, 24x^3 times (-7) is -168x^3. Yes. Wait, but wait, 24x^4 -168x^3: let's factor out 24x^3? Wait, 24x^4 is 24x^3 * x, 168x^3 is 24x^3 * 7? Wait, 24×7=168? Yes! So 24x^4 -168x^3 = 24x^3(x) - 24x^3(7) = 24x^3(x - 7). Wait, but wait, the numerator is 6x. So let's check the GCF of numerator and denominator. The numerator is 6x, denominator is 24x^3(x - 7). So GCF of 6 and 24 is 6, GCF of x and x^3 is x. So we can factor out 6x from numerator and denominator.

Wait, let's do it step by step. The original expression is \(\frac{6x}{24x^4 - 168x^3}\). First, factor the denominator. Let's find the GCF of the two terms in the denominator: 24x^4 and -168x^3. The GCF of 24 and 168 is 24? Wait, no, 24 and 168: 24 is 2^3×3, 168 is 2^3×3×7. So GCF is 2^3×3=24? Wait, no, 24×7=168, so 24 and 168: GCF is 24? Wait, 24 divides into 168 seven times. So yes, GCF of 24 and 168 is 24. And for the variables, \(x^4\) and \(x^3\), GCF is \(x^3\). So denominator factors as \(24x^3(x - 7)\)? Wait, no, 24x^3 * x =24x^4, 24x^3 * (-7)= -168x^3. So yes. Wait, but let's check: 24x^3(x -7)=24x^4 -168x^3. Correct. Now the numerator is 6x. So let's write the expression as \(\frac{6x}{24x^3(x - 7)}\). Now, simplify the coefficients and the variables. The coefficient 6 and 24: 6/24 = 1/4. The variable x and x^3: x/x^3 = 1/x^2. So putting it together: (1/4) * (1/x^2) * (1/(x -7))? Wait, no, wait: numerator is 6x, denominator is 24x^3(x -7). So 6x / (24x^3(x -7)) = (6/24) * (x / x^3) * (1/(x -7)) = (1/4) * (1/x^2) * (1/(x -7)) = 1/(4x^2(x -7)). Wait, but wait, maybe I made a mistake in factoring the denominator. Let's check again. Wait, 24x^4 -168x^3: let's factor out 6x^3 instead? Wait, 6x^3 times 4x is 24x^4, 6x^3 times (-28) is -168x^3? No, 6x^3*4x=24x^4, 6x^3*(-28)= -168x^3? 6*28=168, yes. But 6x^3 is not the GCF. Wait, the GCF of 24 and 168 is 24? Wait, no, 24 and 168: GCF is 24? Wait, 24 divides 168? 168 ÷24=7, so yes. So 24x^4 -168x^3=24x^3(x -7). Then numerator is 6x. So 6x / (24x^3(x -7))= (6/24)*(x/x^3)*(1/(x -7))= (1/4)*(1/x^2)*(1/(x -7))= 1/(4x^2(x -7)). Wait, but let's check with another approach. Let's factor numerator and denominator:

Numerator: 6x = 6 * x

Denominator: 24x^4 -168x^3 = 24x^3(x) - 168x^3 = 24x^3(x) - 24x^3(7) [since 24*7=168] = 24x^3(x -7)

So now, the fraction is (6x) / (24x^3(x -7))

Simplify 6/24 = 1/4, and x/x^3 = 1/x^2 (since x^1 / x^3 = x^(1-3) = x^(-2) = 1/x^2)

So putting it together: (1/4) * (1/x^2) * (1/(x -7)) = 1/(4x^2(x -7))

Wait, but let's check if we can factor the denominator differently. Wait, maybe I made a mistake in the GCF. Let's check the denominator again: 24x^4 -168x^3. Let's factor out 6x^3 instead. 6x^3*4x=24x^4, 6x^3*(-28)= -168x^3? No, 6*28=168, so 6x^3*(-28)= -168x^3. But 6x^3 is not the GCF, because 24 and 168 have a GCF of 24, not 6. Wait, 24 is 6*4, 168 is 6*28. So GCF of 24 and 168 is 6*4? No, GCF is the greatest common factor. So 24 and 168: factors of 24 are 1,2,3,4,6,8,12,24. Factors of 168 are 1,2,3,4,6,7,8,12,14,21,24,28,42,56,84,168. So the greatest common factor is 24. So we should factor out 24x^3. So denominator is 24x^3(x -7). Then numerator is 6x. So 6x / (24x^3(x -7)) = (6/24) * (x/x^3) * (1/(x -7)) = (1/4) * (1/x^2) * (1/(x -7)) = 1/(4x^2(x -7)).

Alternatively, let's do it step by step:

Original expression: \(\frac{6x}{24x^4 - 168x^3}\)

Step 1: Factor the denominator. Find the GCF of \(24x^4\) and \(168x^3\). The GCF of 24 and 168 is 24, and the GCF of \(x^4\) and \(x^3\) is \(x^3\). So:

\(24x^4 - 168x^3 = 24x^3(x) - 24x^3(7) = 24x^3(x - 7)\)

Step 2: Rewrite the expression with the factored denominator:

\(\frac{6x}{24x^3(x - 7)}\)

Step 3: Simplify the coefficients and the variables.

Simplify the coefficient: \(\frac{6}{24} = \frac{1}{4}\)

Simplify the variable: \(\frac{x}{x^3} = \frac{1}{x^2}\) (since \(x = x^1\), and \(x^1 / x^3 = x^{1-3} = x^{-2} = \frac{1}{x^2}\))

Step 4: Multiply the simplified parts together:

\(\frac{1}{4} \times \frac{1}{x^2} \times \frac{1}{x - 7} = \frac{1}{4x^2(x - 7)}\)

So the simplified expression is \(\frac{1}{4x^2(x - 7)}\) or we can expand the denominator: \(\frac{1}{4x^3 - 28x^2}\) but factored form is better.

Answer:

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