Calculate (10/13)^8: Evaluating Powers of Fractions

Q: Why do I need to apply the exponent to both parts?

Because exponent rules state that $ \left(\frac{a}{b}\right)^n = \frac{a^n}{b^n} $. Think of it as multiplying the fraction by itself 8 times - each multiplication affects both top and bottom!

Q: What if I flip the fraction by accident?

Be careful! $ \left(\frac{10}{13}\right)^8 $ gives $ \frac{10^8}{13^8} $, but flipping gives $ \frac{13^8}{10^8} $. Always keep the original order of numerator and denominator.

Q: Do I need to calculate the actual numbers?

Not necessarily! The question asks for the expression form. $ \frac{10^8}{13^8} $ is the correct answer even without computing $ 10^8 = 100,000,000 $.

Q: How is this different from multiplying fractions?

With multiplication, you multiply numerators together and denominators together. With powers, you apply the same exponent to both parts of one fraction.

Q: What if the exponent was negative?

The same rule applies! $ \left(\frac{10}{13}\right)^{-8} = \frac{10^{-8}}{13^{-8}} $, which can also be written as $ \frac{13^8}{10^8} $ using negative exponent rules.

Fraction Powers with Exponent Rules

Insert the corresponding expression:

$\left(\frac{10}{13}\right)^8=$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:08 Let's simplify the following problem.

00:11 Remember, when a fraction is raised to a power, like N, each part, both top and bottom, also gets that power.

00:18 So, we raise both the numerator and the denominator to the power of N.

00:23 Let's apply this rule to our problem.

00:26 And here you have the solution!

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Insert the corresponding expression:

$\left(\frac{10}{13}\right)^8=$

Step-by-step solution

The fraction $\frac{10}{13}$ raised to the power of 8 can be expressed by applying the power to both the numerator and the denominator based on the rule for powers of a fraction:

$\left(\frac{10}{13}\right)^8 = \frac{10^8}{13^8}$

To solve for the given expression, we use the formula $\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$ . This means that the fraction power rule allows us to take each component of the fraction and raise it to the required power:

Step 1: Apply the power of 8 to the numerator: $10^8$ .
Step 2: Apply the power of 8 to the denominator: $13^8$ .
Step 3: Combine both into a single fraction: $\frac{10^8}{13^8}$ .

Thus, the expression $\left(\frac{10}{13}\right)^8$ simplifies to $\frac{10^8}{13^8}$ .

Therefore, the correct answer from the choices provided is $\frac{10^8}{13^8}$ , corresponding to choice 3.

Final Answer

$\frac{10^8}{13^8}$

Key Points to Remember

Essential concepts to master this topic

Rule: Apply the exponent to both numerator and denominator separately
Technique: $\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$ for all fraction powers
Check: Each part should have the same exponent: $\frac{10^8}{13^8}$ ✓

Common Mistakes

Avoid these frequent errors

Applying the exponent to only one part of the fraction
Don't apply 8 to just the numerator getting $\frac{10^8}{13}$ or just the denominator getting $\frac{10}{13^8}$ = wrong answer! The exponent rule breaks down and you get an incorrect expression. Always apply the exponent to both the numerator AND denominator.

Practice Quiz

Test your knowledge with interactive questions

$112^0=\text{?}$

FAQ

Everything you need to know about this question

Why do I need to apply the exponent to both parts?

Because exponent rules state that $\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$ . Think of it as multiplying the fraction by itself 8 times - each multiplication affects both top and bottom!

What if I flip the fraction by accident?

Be careful! $\left(\frac{10}{13}\right)^8$ gives $\frac{10^8}{13^8}$ , but flipping gives $\frac{13^8}{10^8}$ . Always keep the original order of numerator and denominator.

Do I need to calculate the actual numbers?

Not necessarily! The question asks for the expression form. $\frac{10^8}{13^8}$ is the correct answer even without computing $10^8 = 100,000,000$ .

How is this different from multiplying fractions?

With multiplication, you multiply numerators together and denominators together. With powers, you apply the same exponent to both parts of one fraction.

What if the exponent was negative?

The same rule applies! $\left(\frac{10}{13}\right)^{-8} = \frac{10^{-8}}{13^{-8}}$ , which can also be written as $\frac{13^8}{10^8}$ using negative exponent rules.

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