Simplify the Quartic Fraction: x⁴/a⁴ Expression Problem

Q: Why can't I just cancel the 4s in the exponents?

The 4s are exponents, not factors you can cancel! You can only cancel common factors in multiplication. Instead, use the rule $ \frac{x^m}{y^m} = \left(\frac{x}{y}\right)^m $ when exponents match.

Q: How do I know when to use this exponent rule?

Look for matching exponents in the numerator and denominator! When you see $ \frac{a^n}{b^n} $, you can always rewrite it as $ \left(\frac{a}{b}\right)^n $.

Q: What if the exponents were different numbers?

If exponents don't match (like $ \frac{x^4}{a^3} $), you cannot use this rule. The expression would stay as is, or you'd need different techniques depending on the problem.

Q: Is there a way to check if my answer is right?

Yes! Expand your answer back using exponent rules. For example: $ \left(\frac{x}{a}\right)^4 = \frac{x^4}{a^4} $ should give you the original expression.

Q: Can this rule work backwards too?

Absolutely! You can go both directions: $ \frac{x^4}{a^4} = \left(\frac{x}{a}\right)^4 $ and $ \left(\frac{x}{a}\right)^4 = \frac{x^4}{a^4} $. Choose whichever form makes your problem easier to solve!

Exponent Rules with Fractional Bases

Insert the corresponding expression:

$\frac{x^4}{a^4}=$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Simplify the following problem

00:03 According to the laws of exponents, a fraction raised to the power (N)

00:06 equals a fraction where both the numerator and denominator are raised to the power (N)

00:09 We will apply this formula to our exercise, in the opposite direction

00:16 We will place the entire fraction inside of parentheses and raise it to the appropriate power

00:20 This is the solution

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Insert the corresponding expression:

$\frac{x^4}{a^4}=$

Step-by-step solution

To solve this problem, we'll follow these steps:

Step 1: Identify the given information: We have the expression $\frac{x^4}{a^4}$ .
Step 2: Apply the appropriate exponent rule: Use the rule $\frac{x^m}{y^m} = \left(\frac{x}{y}\right)^m$ .
Step 3: Simplify the expression using the rule: Substitute $m$ with 4, $x$ with $x$ , and $y$ with $a$ .

Now, let's work through each step:
Step 1: The expression is $\frac{x^4}{a^4}$ .
Step 2: We use the rule $\frac{x^m}{y^m} = \left(\frac{x}{y}\right)^m$ , which states that a fraction where the numerator and denominator are raised to the same power can be expressed as a power of a single fraction.
Step 3: Plugging in the values, we have $\frac{x^4}{a^4} = \left(\frac{x}{a}\right)^4$ .

Therefore, the solution to the problem is $\left(\frac{x}{a}\right)^4$ , which corresponds to choice 1.

Final Answer

$\left(\frac{x}{a}\right)^4$

Key Points to Remember

Essential concepts to master this topic

Power Rule: Same exponents allow factoring into single fraction raised to power
Technique: $\frac{x^4}{a^4} = \left(\frac{x}{a}\right)^4$ using quotient rule
Check: Expand back: $\left(\frac{x}{a}\right)^4 = \frac{x^4}{a^4}$ ✓

Common Mistakes

Avoid these frequent errors

Treating numerator and denominator exponents separately
Don't simplify $\frac{x^4}{a^4}$ as $\frac{x^4}{4a}$ by moving the 4! This ignores the exponent rule and creates a completely different expression. Always recognize when numerator and denominator have the same exponent and factor as a single fraction raised to that power.

Practice Quiz

Test your knowledge with interactive questions

$$ Choose the corresponding expression:

$\left(\frac{1}{2}\right)^2=$

FAQ

Everything you need to know about this question

Why can't I just cancel the 4s in the exponents?

The 4s are exponents, not factors you can cancel! You can only cancel common factors in multiplication. Instead, use the rule $\frac{x^m}{y^m} = \left(\frac{x}{y}\right)^m$ when exponents match.

How do I know when to use this exponent rule?

Look for matching exponents in the numerator and denominator! When you see $\frac{a^n}{b^n}$ , you can always rewrite it as $\left(\frac{a}{b}\right)^n$ .

What if the exponents were different numbers?

If exponents don't match (like $\frac{x^4}{a^3}$ ), you cannot use this rule. The expression would stay as is, or you'd need different techniques depending on the problem.

Is there a way to check if my answer is right?

Yes! Expand your answer back using exponent rules. For example: $\left(\frac{x}{a}\right)^4 = \frac{x^4}{a^4}$ should give you the original expression.

Can this rule work backwards too?

Absolutely! You can go both directions: $\frac{x^4}{a^4} = \left(\frac{x}{a}\right)^4$ and $\left(\frac{x}{a}\right)^4 = \frac{x^4}{a^4}$ . Choose whichever form makes your problem easier to solve!

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