Calculate (13/19)^7: Evaluating a Fraction Raised to the Seventh Power

Q: What's the difference between multiplying by the exponent and raising to the exponent?

Multiplying gives $ \frac{13 \times 7}{19 \times 7} = \frac{91}{133} $, but raising to the power gives $ \frac{13^7}{19^7} $. These are completely different operations! Exponentiation means repeated multiplication, not just multiplying by the exponent number.

Q: Can I apply this rule to any fraction raised to any power?

Yes! The rule $ \left(\frac{a}{b}\right)^n = \frac{a^n}{b^n} $ works for any fraction and any positive integer exponent. Just remember to apply the exponent to both the top and bottom numbers.

Q: What if I accidentally apply the exponent to only the numerator or denominator?

You'll get a wrong answer! For example, $ \frac{13^7}{19} $ or $ \frac{13}{19^7} $ are both incorrect. Always remember: both parts of the fraction must have the same exponent when you raise the entire fraction to a power.

Fractional Exponents with Power Rule

Insert the corresponding expression:

$\left(\frac{13}{19}\right)^7=$

❤️ Continue Your Math Journey!

We have hundreds of course questions with personalized recommendations + Account 100% premium

Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:07 Let's simplify this problem step by step.

00:11 Remember, if you raise a fraction to a power, it means both the top and the bottom numbers are also raised to that power.

00:19 So, let's apply this rule to our example. Take a deep breath, we've got this!

00:24 Here's the part where we solve it together, using what we've just learned.

00:29 And there you have it! That's the solution. Great job!

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Insert the corresponding expression:

$\left(\frac{13}{19}\right)^7=$

Step-by-step solution

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

Step 1: Identify the given expression $\left(\frac{13}{19}\right)^7$ .
Step 2: Apply the exponent rule for fractions: $\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$ .
Step 3: Perform the calculation by raising both the numerator and the denominator to the power of 7.

Now, let's work through each step:
Step 1: The expression provided is $\left(\frac{13}{19}\right)^7$ , which is a fraction raised to an exponent.
Step 2: Using the exponentiation rule for fractions: $\left(\frac{a}{b}\right)^n$ is equivalent to $\frac{a^n}{b^n}$ .
Step 3: Applying this rule, we express $\left(\frac{13}{19}\right)^7$ as $\frac{13^7}{19^7}$ .

Therefore, the solution to the problem is $\frac{13^7}{19^7}$ , which corresponds to choice 1.

Final Answer

$\frac{13^7}{19^7}$

Key Points to Remember

Essential concepts to master this topic

Power Rule: When raising fractions to exponents, apply power to both numerator and denominator
Technique: $\left(\frac{13}{19}\right)^7 = \frac{13^7}{19^7}$ using the rule $\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$
Check: Verify both parts have same exponent: $\frac{13^7}{19^7}$ has 7 in numerator and denominator ✓

Common Mistakes

Avoid these frequent errors

Multiplying the exponent instead of applying the power rule
Don't change $\left(\frac{13}{19}\right)^7$ to $\frac{13 \times 7}{19 \times 7}$ = $\frac{91}{133}$ ! This multiplies instead of raising to a power, giving a completely wrong result. Always apply the exponent to both numerator and denominator separately using $\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$ .

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 does the exponent apply to both the numerator and denominator?

When you raise a fraction to a power, you're multiplying that fraction by itself multiple times. Since $\left(\frac{13}{19}\right)^7 = \frac{13}{19} \times \frac{13}{19} \times \frac{13}{19} \times \frac{13}{19} \times \frac{13}{19} \times \frac{13}{19} \times \frac{13}{19}$ , this gives you 13 multiplied by itself 7 times in the numerator and 19 multiplied by itself 7 times in the denominator.

What's the difference between multiplying by the exponent and raising to the exponent?

Multiplying gives $\frac{13 \times 7}{19 \times 7} = \frac{91}{133}$ , but raising to the power gives $\frac{13^7}{19^7}$ . These are completely different operations! Exponentiation means repeated multiplication, not just multiplying by the exponent number.

Do I need to calculate the actual values of 13^7 and 19^7?

Not usually! Most problems ask you to express the answer in exponential form like $\frac{13^7}{19^7}$ . This form is often preferred because the actual numbers (like 13^7 = 62,748,517) are very large and difficult to work with.

Can I apply this rule to any fraction raised to any power?

Yes! The rule $\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$ works for any fraction and any positive integer exponent. Just remember to apply the exponent to both the top and bottom numbers.

What if I accidentally apply the exponent to only the numerator or denominator?

You'll get a wrong answer! For example, $\frac{13^7}{19}$ or $\frac{13}{19^7}$ are both incorrect. Always remember: both parts of the fraction must have the same exponent when you raise the entire fraction to a power.

🌟 Unlock Your Math Potential

Get unlimited access to all 18 Exponents Rules questions, detailed video solutions, and personalized progress tracking.

📹

Unlimited Video Solutions

Step-by-step explanations for every problem

📊

Progress Analytics

Track your mastery across all topics

🚫

Ad-Free Learning

Focus on math without distractions

No credit card required • Cancel anytime