Simplify the Expression: 4^6 Divided by (a^6 × x^6)

Q: Why can't I just calculate 4^6 first and divide by each term?

While mathematically possible, it's much more complex and doesn't show the elegant pattern! The quotient power rule $ \frac{a^n}{b^n} = \left(\frac{a}{b}\right)^n $ gives a cleaner, simplified form.

Q: How do I know when to use the quotient power rule?

Look for matching exponents in the numerator and denominator! When you see the same power (like 6) on all terms, you can factor it out using $ \left(\frac{base}{base}\right)^{power} $.

Q: Is my answer still correct if I leave it as the original fraction?

Yes, but simplified form is preferred! $ \left(\frac{4}{a \times x}\right)^6 $ is cleaner and shows you understand exponent rules. It's like reducing fractions - both are correct, but simplified is better.

Q: What if the exponents were different numbers?

Then you cannot use this rule! The quotient power rule only works when all terms have identical exponents. Different exponents require different approaches.

Q: Can I write the denominator as (ax)^6 instead of (a×x)^6?

Yes! Both $ (a \times x)^6 $ and $ (ax)^6 $ mean the same thing. The multiplication symbol is often omitted in algebra, so both forms are perfectly correct.

Exponent Rules with Quotient Powers

Insert the corresponding expression:

$\frac{4^6}{a^6\times x^6}=$

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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 product that is raised to the power (N)

00:07 is equal to a product broken down into factors where each factor is raised to the power (N)

00:13 We will apply this formula to our exercise, in the reverse direction

00:17 We'll combine the factors into a multiplication operation within parentheses

00:22 According to the laws of exponents, a fraction that is raised to the power (N)

00:26 equals a fraction where both the numerator and the denominator are raised to the power (N)

00:29 We will apply this formula to our exercise, in the reverse direction

00:32 We'll place the entire fraction inside parentheses and raise it to the appropriate power

00:36 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:

$\frac{4^6}{a^6\times x^6}=$

Step-by-step solution

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

Step 1: Recognize the given expression $\frac{4^6}{a^6 \times x^6}$ is a power over a product raised to that power.
Step 2: Apply the power of a quotient rule to rewrite the expression.

Now, let's work through each step:
Step 1: The given expression is $\frac{4^6}{a^6 \times x^6}$ . This can be seen as having common exponents across the numerator and the denominator.
Step 2: Using the power of a quotient rule, which allows us to express the initial expression as $\left(\frac{4}{a \times x}\right)^6$ . This step involves recognizing that you can treat the entire $a \times x$ as a single base for the denominator.

Hence, the simplified form of the given expression is $\left(\frac{4}{a \times x}\right)^6$ .

Therefore, the solution to the problem is $\left(\frac{4}{a \times x}\right)^6$ .

Final Answer

$\left(\frac{4}{a\times x}\right)^6$

Key Points to Remember

Essential concepts to master this topic

Rule: When dividing powers with same exponents, factor out the common exponent
Technique: Transform $\frac{4^6}{a^6 \times x^6}$ into $\left(\frac{4}{a \times x}\right)^6$
Check: Expand your answer back to verify it equals the original expression ✓

Common Mistakes

Avoid these frequent errors

Trying to combine bases with different variables
Don't write $\frac{4}{(ax)^6}$ or calculate 4^6 ÷ a^6 ÷ x^6 separately = wrong form! This ignores that all terms have the same exponent. Always recognize the pattern and factor out the common exponent using quotient power rules.

Practice Quiz

Test your knowledge with interactive questions

$112^0=\text{?}$

FAQ

Everything you need to know about this question

Why can't I just calculate 4^6 first and divide by each term?

While mathematically possible, it's much more complex and doesn't show the elegant pattern! The quotient power rule $\frac{a^n}{b^n} = \left(\frac{a}{b}\right)^n$ gives a cleaner, simplified form.

How do I know when to use the quotient power rule?

Look for matching exponents in the numerator and denominator! When you see the same power (like 6) on all terms, you can factor it out using $\left(\frac{base}{base}\right)^{power}$ .

Is my answer still correct if I leave it as the original fraction?

Yes, but simplified form is preferred! $\left(\frac{4}{a \times x}\right)^6$ is cleaner and shows you understand exponent rules. It's like reducing fractions - both are correct, but simplified is better.

What if the exponents were different numbers?

Then you cannot use this rule! The quotient power rule only works when all terms have identical exponents. Different exponents require different approaches.

Can I write the denominator as (ax)^6 instead of (a×x)^6?

Yes! Both $(a \times x)^6$ and $(ax)^6$ mean the same thing. The multiplication symbol is often omitted in algebra, so both forms are perfectly correct.

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