Simplify the Rational Expression: (8x²-1)/(24x⁵-3x³)

Q: Why can't I just cancel the 8x² from top and bottom?

You can only cancel factors, not terms! The $ 8x^2 $ is part of $ 8x^2-1 $ (a term), while in the denominator it's mixed with other terms. You must factor completely first.

Q: How do I know what to factor out of the denominator?

Look for the greatest common factor (GCF)! In $ 24x^5-3x^3 $, both terms have $ 3x^3 $ in common. Factor this out: $ 3x^3(8x^2-1) $.

Q: What if the numerator and denominator don't have common factors?

Then the rational expression is already in simplest form! Not every rational expression can be simplified further. Always factor first to check.

Q: Can I factor the numerator 8x²-1 further?

Yes! $ 8x^2-1 $ is a difference of squares: $ (2\sqrt{2}x)^2-1^2 $, but this doesn't help with cancellation in this problem since the factored form doesn't appear in the denominator.

Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Simply

00:06 Let's break down 24 into factors 8 and 3

00:11 Let's break down power of 5 into power of 3 and square

00:20 Let's mark the common factors

00:41 Let's take out the common factors from the parentheses

00:56 Let's reduce what we can

01:01 And this is the solution to the question

Step-by-step written solution

Follow each step carefully to understand the complete solution

1

Understand the problem

Simplify:

$\frac{8x^2-1}{24x^5-3x^3}$

2

Step-by-step solution

Let's simplify the given expression:

$\frac{8x^2-1}{24x^5-3x^3}$ Remember that we can reduce complete expressions only when both the numerator and denominator are completely factored into multiplication expressions,

For this, we'll use factorization, identify where in the denominator we can factor out a common factor, do this, then reduce the expressions possible in the fraction we get (reduction sign):

$\frac{8x^2-1}{24x^5-3x^3} \\ \frac{8x^2-1}{3x^3(8x^2-1)} \\ \downarrow\\ \boxed{\frac{1}{3x^3}}$ In the first stage, to factor out the common factor in the denominator, we also used the law of exponents: $a^m\cdot a^n=a^{m+n}$

Therefore, the correct answer is answer D.

3

Final Answer

$\frac{1}{3x^3}$

Key Points to Remember

Essential concepts to master this topic

Rule: Factor both numerator and denominator completely before reducing
Technique: Factor out $3x^3$ from denominator: $24x^5-3x^3 = 3x^3(8x^2-1)$
Check: Verify $\frac{1}{3x^3} \cdot 3x^3(8x^2-1) = 8x^2-1$ ✓

Common Mistakes

Avoid these frequent errors

Canceling terms instead of factors
Don't cancel individual terms like canceling $8x^2$ from numerator with part of denominator = wrong answer! Terms can only be canceled when they appear as complete factors in both numerator and denominator. Always factor completely first, then cancel common factors.

Practice Quiz

Test your knowledge with interactive questions

Determine if the simplification shown below is correct:

$\frac{7}{7\cdot8}=8$

FAQ

Everything you need to know about this question

Why can't I just cancel the 8x² from top and bottom?

+

You can only cancel factors, not terms! The $8x^2$ is part of $8x^2-1$ (a term), while in the denominator it's mixed with other terms. You must factor completely first.

How do I know what to factor out of the denominator?

+

Look for the greatest common factor (GCF)! In $24x^5-3x^3$ , both terms have $3x^3$ in common. Factor this out: $3x^3(8x^2-1)$ .

What if the numerator and denominator don't have common factors?

+

Then the rational expression is already in simplest form! Not every rational expression can be simplified further. Always factor first to check.

Can I factor the numerator 8x²-1 further?

+

Yes! $8x^2-1$ is a difference of squares: $(2\sqrt{2}x)^2-1^2$ , but this doesn't help with cancellation in this problem since the factored form doesn't appear in the denominator.

How can I check if my simplified answer is correct?

+

Multiply your answer by what you canceled out. If you get the original expression back, you're right! Here: $\frac{1}{3x^3} \times 3x^3(8x^2-1) = 8x^2-1$ ✓

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Simplify the Rational Expression: (8x²-1)/(24x⁵-3x³)

Rational Expression Simplification with Factorization

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