Solve (10/13)^(-4): Negative Exponent Calculation

Q: Why don't we flip the fraction when we have a negative exponent?

While $ \left(\frac{10}{13}\right)^{-4} $ does equal $ \left(\frac{13}{10}\right)^4 $, the question asks for a specific form. We need to apply the negative exponent rule while keeping the original fraction structure.

Q: What's the difference between the answer choices?

The key difference is where the negative exponent appears:$ \frac{10^{-4}}{13^{-4}} $ applies it to both parts (correct)$ \frac{10^{-4}}{13} $ applies it only to numerator (incorrect)$ \frac{10}{13^{-4}} $ applies it only to denominator (incorrect)

Q: How do I remember the negative exponent rule for fractions?

Think of it as distributing the exponent! Just like $ (ab)^n = a^n b^n $, we have $ \left(\frac{a}{b}\right)^n = \frac{a^n}{b^n} $. The exponent affects every part of the base.

Q: Are negative exponents always confusing like this?

Not at all! Once you understand that negative exponents follow the same distribution rules as positive ones, they become much clearer. Practice with simpler examples first, like $ \left(\frac{1}{2}\right)^{-2} = \frac{1^{-2}}{2^{-2}} $.

Q: Can I simplify this expression further?

Yes! $ \frac{10^{-4}}{13^{-4}} $ can be rewritten as $ \frac{1/10^4}{1/13^4} = \frac{13^4}{10^4} = \left(\frac{13}{10}\right)^4 $, but the question asks for the specific form with separated negative exponents.

Negative Exponents with Fraction Bases

Insert the corresponding expression:

$\left(\frac{10}{13}\right)^{-4}=$

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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)

00:09 equals the numerator and denominator, each raised to the same power (N)

00:13 We will apply this formula to our exercise

00:17 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:

$\left(\frac{10}{13}\right)^{-4}=$

Step-by-step solution

To solve the expression $\left(\frac{10}{13}\right)^{-4}$ , we start by applying the rule for dividing exponents is:

$\frac{10^{-4}}{13^{-4}}$ , which maintains the negative exponent but as separate components of fraction resulting in the same value.

Consequently, the expression $\left(\frac{10}{13}\right)^{-4}$ equates to $\frac{10^{-4}}{13^{-4}}$ .

By comparing this with the presented choices, we identify that option (2):

$\frac{10^{-4}}{13^{-4}}$

matches correctly with our conversion of the original expression.

Therefore, the correct expression is $\frac{10^{-4}}{13^{-4}}$ .

Final Answer

$\frac{10^{-4}}{13^{-4}}$

Key Points to Remember

Essential concepts to master this topic

Rule: Apply negative exponent to numerator and denominator separately
Technique: $\left(\frac{a}{b}\right)^{-n} = \frac{a^{-n}}{b^{-n}}$ keeps the fraction structure
Check: Both forms equal $\left(\frac{13}{10}\right)^4$ when simplified ✓

Common Mistakes

Avoid these frequent errors

Applying negative exponent incorrectly to fractions
Don't flip the fraction first like $\left(\frac{10}{13}\right)^{-4} = \left(\frac{13}{10}\right)^4$ = wrong format! This changes the required expression form. Always keep the original fraction structure and apply the negative exponent to both numerator and denominator: $\frac{10^{-4}}{13^{-4}}$ .

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 don't we flip the fraction when we have a negative exponent?

While $\left(\frac{10}{13}\right)^{-4}$ does equal $\left(\frac{13}{10}\right)^4$ , the question asks for a specific form. We need to apply the negative exponent rule while keeping the original fraction structure.

What's the difference between the answer choices?

The key difference is where the negative exponent appears:

$\frac{10^{-4}}{13^{-4}}$ applies it to both parts (correct)
$\frac{10^{-4}}{13}$ applies it only to numerator (incorrect)
$\frac{10}{13^{-4}}$ applies it only to denominator (incorrect)

How do I remember the negative exponent rule for fractions?

Think of it as distributing the exponent! Just like $(ab)^n = a^n b^n$ , we have $\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$ . The exponent affects every part of the base.

Are negative exponents always confusing like this?

Not at all! Once you understand that negative exponents follow the same distribution rules as positive ones, they become much clearer. Practice with simpler examples first, like $\left(\frac{1}{2}\right)^{-2} = \frac{1^{-2}}{2^{-2}}$ .

Can I simplify this expression further?

Yes! $\frac{10^{-4}}{13^{-4}}$ can be rewritten as $\frac{1/10^4}{1/13^4} = \frac{13^4}{10^4} = \left(\frac{13}{10}\right)^4$ , but the question asks for the specific form with separated negative exponents.

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