Simplify the Expression: Evaluating 3^6/8^6 Step-by-Step

Q: Why can I move the exponent outside the fraction?

Because of the quotient rule for exponents! When you have the same exponent on both numerator and denominator, $ \frac{a^n}{b^n} = \left(\frac{a}{b}\right)^n $. It's like distributing the exponent to each part.

Q: What if the exponents were different?

Then you cannot use this rule! For example, $ \frac{3^4}{8^6} $ doesn't simplify to $ \left(\frac{3}{8}\right)^{something} $. The exponents must be identical to apply this rule.

Q: Should I calculate the final numerical answer?

Not necessarily! The expression $ \left(\frac{3}{8}\right)^6 $ is often the preferred form because it's simpler and shows the mathematical structure clearly. Calculate only if specifically asked.

Q: Can this work with variables too?

Absolutely! For example, $ \frac{x^5}{y^5} = \left(\frac{x}{y}\right)^5 $. The quotient rule works with any bases as long as they're non-zero and have the same exponent.

Q: What if I see this written backwards?

The rule works both ways! You can go from $ \left(\frac{a}{b}\right)^n $ to $ \frac{a^n}{b^n} $ or vice versa. Choose whichever form makes your problem easier to solve.

Exponent Rules with Quotient Powers

Insert the corresponding expression:

$\frac{3^6}{8^6}=$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Simplify the following problem

00:04 According to the laws of exponents, a fraction raised to the power (N)

00:08 equals the numerator and denominator, each raised to the same power (N)

00:12 We'll apply this formula to our exercise, only this time in the opposite direction

00:21 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:

$\frac{3^6}{8^6}=$

Step-by-step solution

To solve this problem, we will apply the rule of exponentiation for fractions. This rule states that $\frac{a^n}{b^n} = \left(\frac{a}{b}\right)^n$ , where $a$ and $b$ are non-zero numbers and $n$ is an integer.

Let's go through the solution step-by-step:

Step 1: Recognize that the expression we need to rewrite is $\frac{3^6}{8^6}$ .
Step 2: Apply the power of a fraction rule. According to this rule, $\frac{3^6}{8^6} = \left(\frac{3}{8}\right)^6$ .
Step 3: Thus, the expression $\frac{3^6}{8^6}$ simplifies to $\left(\frac{3}{8}\right)^6$ .

The solution to the problem is that the expression can be rewritten as $\left(\frac{3}{8}\right)^6$ .

Final Answer

$\left(\frac{3}{8}\right)^6$

Key Points to Remember

Essential concepts to master this topic

Rule: $\frac{a^n}{b^n} = \left(\frac{a}{b}\right)^n$ when bases have same exponents
Technique: Combine bases first: $\frac{3^6}{8^6} = \left(\frac{3}{8}\right)^6$
Check: Both sides equal $\frac{729}{262144}$ when calculated ✓

Common Mistakes

Avoid these frequent errors

Trying to simplify exponents before combining bases
Don't calculate 3^6 = 729 and 8^6 = 262,144 first = massive numbers that are hard to work with! This makes the problem unnecessarily complicated and error-prone. Always apply the quotient rule first to get $\left(\frac{3}{8}\right)^6$ , which is much simpler.

Practice Quiz

Test your knowledge with interactive questions

$112^0=\text{?}$

FAQ

Everything you need to know about this question

Why can I move the exponent outside the fraction?

Because of the quotient rule for exponents! When you have the same exponent on both numerator and denominator, $\frac{a^n}{b^n} = \left(\frac{a}{b}\right)^n$ . It's like distributing the exponent to each part.

What if the exponents were different?

Then you cannot use this rule! For example, $\frac{3^4}{8^6}$ doesn't simplify to $\left(\frac{3}{8}\right)^{something}$ . The exponents must be identical to apply this rule.

Should I calculate the final numerical answer?

Not necessarily! The expression $\left(\frac{3}{8}\right)^6$ is often the preferred form because it's simpler and shows the mathematical structure clearly. Calculate only if specifically asked.

Can this work with variables too?

Absolutely! For example, $\frac{x^5}{y^5} = \left(\frac{x}{y}\right)^5$ . The quotient rule works with any bases as long as they're non-zero and have the same exponent.

What if I see this written backwards?

The rule works both ways! You can go from $\left(\frac{a}{b}\right)^n$ to $\frac{a^n}{b^n}$ or vice versa. Choose whichever form makes your problem easier to solve.

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