Question

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

Explanation:

Step1: Factor numerator and denominator

First, factor the numerator \(6x\) (it's already factored). For the denominator \(24x^{4}-168x^{3}\), we can factor out the greatest common factor (GCF). 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 \(2^3\times3\), 168 is \(2^3\times3\times7\), so GCF is \(2^3\times3 = 24\)? Wait, no, 24 and 168: 168 ÷ 24 = 7, so GCF is 24? Wait, no, 24x^4 and 168x^3: the GCF for the coefficients is 24? Wait, 24 and 168: 247=168, so GCF is 24? Wait, no, 24 and 168: GCF is 24? Wait, 247=168, so GCF is 24? Wait, no, 24 is 24, 168 is 247, so GCF is 24. And for the variables, \(x^4\) and \(x^3\), GCF is \(x^3\). So the GCF of \(24x^4\) and \(168x^3\) is \(24x^3\)? Wait, no, 24x^4 = 24x^3 x, 168x^3 = 24x^3 7. Wait, 247=168? No, 247=168? 247=168, yes. Wait, 24x^4 - 168x^3 = 24x^3(x) - 24x^3(7) = 24x^3(x - 7)? Wait, no, 24x^4 is 24x^3 x, 168x^3 is 24x^3 7? Wait, 247=168, yes. Wait, but 24x^4 - 168x^3 = 24x^3(x) - 168x^3 = x^3(24x - 168) = x^324(x - 7) = 24x^3(x - 7). Wait, but let's check the original denominator: 24x^4 - 168x^3. Let's factor out 24x^3? Wait, 24x^4 ÷ 24x^3 = x, 168x^3 ÷ 24x^3 = 7. So denominator becomes 24x^3(x - 7). Wait, but 24x^3(x - 7) = 24x^4 - 168x^3, which is correct. Wait, but also, we can factor out 12x^3? Wait, 24x^4 - 168x^3 = 12x^3(2x - 14) = 12x^32(x - 7) = 24x^3(x - 7), same result. Alternatively, factor out 6x^3: 24x^4 - 168x^3 = 6x^3(4x - 28) = 6x^34(x - 7) = 24x^3(x - 7). So denominator factors to 24x^3(x - 7), numerator is 6x. Wait, but let's do it step by step.

Wait, maybe a better way: factor denominator by GCF. Let's find GCF of 24 and 168: GCF(24,168)=24? Wait, 24*7=168, so yes, GCF is 24. And GCF of x^4 and x^3 is x[LLM SSE On Failure]

Answer:

Step1: Factor numerator and denominator

First, factor the numerator \(6x\) (it's already factored). For the denominator \(24x^{4}-168x^{3}\), we can factor out the greatest common factor (GCF). 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 \(2^3\times3\), 168 is \(2^3\times3\times7\), so GCF is \(2^3\times3 = 24\)? Wait, no, 24 and 168: 168 ÷ 24 = 7, so GCF is 24? Wait, no, 24x^4 and 168x^3: the GCF for the coefficients is 24? Wait, 24 and 168: 247=168, so GCF is 24? Wait, no, 24 and 168: GCF is 24? Wait, 247=168, so GCF is 24? Wait, no, 24 is 24, 168 is 247, so GCF is 24. And for the variables, \(x^4\) and \(x^3\), GCF is \(x^3\). So the GCF of \(24x^4\) and \(168x^3\) is \(24x^3\)? Wait, no, 24x^4 = 24x^3 x, 168x^3 = 24x^3 7. Wait, 247=168? No, 247=168? 247=168, yes. Wait, 24x^4 - 168x^3 = 24x^3(x) - 24x^3(7) = 24x^3(x - 7)? Wait, no, 24x^4 is 24x^3 x, 168x^3 is 24x^3 7? Wait, 247=168, yes. Wait, but 24x^4 - 168x^3 = 24x^3(x) - 168x^3 = x^3(24x - 168) = x^324(x - 7) = 24x^3(x - 7). Wait, but let's check the original denominator: 24x^4 - 168x^3. Let's factor out 24x^3? Wait, 24x^4 ÷ 24x^3 = x, 168x^3 ÷ 24x^3 = 7. So denominator becomes 24x^3(x - 7). Wait, but 24x^3(x - 7) = 24x^4 - 168x^3, which is correct. Wait, but also, we can factor out 12x^3? Wait, 24x^4 - 168x^3 = 12x^3(2x - 14) = 12x^32(x - 7) = 24x^3(x - 7), same result. Alternatively, factor out 6x^3: 24x^4 - 168x^3 = 6x^3(4x - 28) = 6x^34(x - 7) = 24x^3(x - 7). So denominator factors to 24x^3(x - 7), numerator is 6x. Wait, but let's do it step by step.

Wait, maybe a better way: factor denominator by GCF. Let's find GCF of 24 and 168: GCF(24,168)=24? Wait, 24*7=168, so yes, GCF is 24. And GCF of x^4 and x^3 is x[LLM SSE On Failure]