Calculate (3/7)^6: Evaluating the Sixth Power of a Fraction

Q: Why do both the 3 and 7 get raised to the 6th power?

Because when you raise a fraction to a power, the exponent applies to the entire fraction. Think of it like this: $ \left(\frac{3}{7}\right)^6 $ means multiply the fraction by itself 6 times, so both parts need the exponent.

Q: What's the difference between $ \frac{3}{7^6} $ and $ \frac{3^6}{7^6} $ ?

Huge difference! $ \frac{3}{7^6} $ means only the denominator is raised to the 6th power, while $ \frac{3^6}{7^6} $ means both parts are raised to the 6th power. Only the second one equals $ \left(\frac{3}{7}\right)^6 $.

Q: Can I calculate the actual numbers instead of leaving it as powers?

Absolutely! $ 3^6 = 729 $ and $ 7^6 = 117,649 $, so the answer could be written as $ \frac{729}{117,649} $. But leaving it in exponential form is usually preferred and easier to work with.

Q: Does this rule work for any fraction and any exponent?

Yes! The rule $ \left(\frac{a}{b}\right)^n = \frac{a^n}{b^n} $ works for any numbers a, b, and n (where b ≠ 0). This is one of the fundamental power rules in mathematics.

Q: What if the exponent was negative, like $ \left(\frac{3}{7}\right)^{-6} $ ?

Great question! With a negative exponent, you flip the fraction first, then apply the positive exponent: $ \left(\frac{3}{7}\right)^{-6} = \left(\frac{7}{3}\right)^6 = \frac{7^6}{3^6} $.

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 a power (N)

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

00:12 We will apply this formula to our exercise

00:16 This is the solution

Step-by-step written solution

Follow each step carefully to understand the complete solution

1

Understand the problem

Insert the corresponding expression:

$\left(\frac{3}{7}\right)^6=$

2

Step-by-step solution

The problem asks us to express $\left(\frac{3}{7}\right)^6$ in another form. To solve this, we apply the exponent rule for fractions: $\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$ .

First, identify the numerator and the denominator in the fraction $\frac{3}{7}$ .
We have $a = 3$ and $b = 7$ .
According to the exponent rule, raise both the numerator and the denominator separately to the power of 6:

$\left(\frac{3}{7}\right)^6 = \frac{3^6}{7^6}$

This signifies that each component of the fraction is raised to the power of 6.

To verify, we compare our result with the given choices:

Option 1: $\frac{3}{7^6}$ does not apply the exponent to the "3".
Option 2: $\frac{3^6}{7^6}$ , matches our derived expression.
Option 3: $\frac{3^6}{7}$ does not apply the exponent to the "7".
Option 4: $6\times\left(\frac{3}{7}\right)^5$ changes the power on the entire fraction and multiplies by 6, which is incorrect based on our interpretation.

Therefore, the solution to the problem is $\frac{3^6}{7^6}$ , which corresponds to choice 2.

3

Final Answer

$\frac{3^6}{7^6}$

Key Points to Remember

Essential concepts to master this topic

Rule: Apply exponent to both numerator and denominator separately
Technique: $\left(\frac{3}{7}\right)^6 = \frac{3^6}{7^6}$ using power rule
Check: Both parts of fraction have same exponent: 6 and 6 ✓

Common Mistakes

Avoid these frequent errors

Applying exponent to only numerator or denominator
Don't write $\frac{3}{7^6}$ or $\frac{3^6}{7}$ = wrong fractional powers! This violates the exponent rule and gives incorrect results. Always apply the exponent to both the numerator AND denominator: $\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$ .

Practice Quiz

Test your knowledge with interactive questions

$112^0=\text{?}$

FAQ

Everything you need to know about this question

Why do both the 3 and 7 get raised to the 6th power?

+

Because when you raise a fraction to a power, the exponent applies to the entire fraction. Think of it like this: $\left(\frac{3}{7}\right)^6$ means multiply the fraction by itself 6 times, so both parts need the exponent.

What's the difference between $\frac{3}{7^6}$ and $\frac{3^6}{7^6}$ ?

+

Huge difference! $\frac{3}{7^6}$ means only the denominator is raised to the 6th power, while $\frac{3^6}{7^6}$ means both parts are raised to the 6th power. Only the second one equals $\left(\frac{3}{7}\right)^6$ .

Can I calculate the actual numbers instead of leaving it as powers?

+

Absolutely! $3^6 = 729$ and $7^6 = 117,649$ , so the answer could be written as $\frac{729}{117,649}$ . But leaving it in exponential form is usually preferred and easier to work with.

Does this rule work for any fraction and any exponent?

+

Yes! The rule $\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$ works for any numbers a, b, and n (where b ≠ 0). This is one of the fundamental power rules in mathematics.

What if the exponent was negative, like $\left(\frac{3}{7}\right)^{-6}$ ?

+

Great question! With a negative exponent, you flip the fraction first, then apply the positive exponent: $\left(\frac{3}{7}\right)^{-6} = \left(\frac{7}{3}\right)^6 = \frac{7^6}{3^6}$ .

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Calculate (3/7)^6: Evaluating the Sixth Power of a Fraction

Exponent Rules with Fractional Bases

❤️ Continue Your Math Journey!

Step-by-step video solution

Step-by-step written solution

Understand the problem

Step-by-step solution

Final Answer

Key Points to Remember

Common Mistakes

Practice Quiz

FAQ

Why do both the 3 and 7 get raised to the 6th power?

What's the difference between $\frac{3}{7^6}$ and $\frac{3^6}{7^6}$ ?

Can I calculate the actual numbers instead of leaving it as powers?

Does this rule work for any fraction and any exponent?

What if the exponent was negative, like $\left(\frac{3}{7}\right)^{-6}$ ?

🌟 Unlock Your Math Potential

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Calculate (3/7)^6: Evaluating the Sixth Power of a Fraction

Exponent Rules with Fractional Bases

❤️ Continue Your Math Journey!

Step-by-step video solution

Step-by-step written solution

Understand the problem

Step-by-step solution

Final Answer

Key Points to Remember

Common Mistakes

Practice Quiz

FAQ

Why do both the 3 and 7 get raised to the 6th power?

What's the difference between 376 \frac{3}{7^6} 763​ and 3676 \frac{3^6}{7^6} 7636​?

Can I calculate the actual numbers instead of leaving it as powers?

Does this rule work for any fraction and any exponent?

What if the exponent was negative, like (37)−6 \left(\frac{3}{7}\right)^{-6} (73​)−6?

🌟 Unlock Your Math Potential

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Powers of a Fraction

Calculate (20/21)⁴: Evaluating the Fourth Power of a Fraction

Calculate (4/7)³: Evaluating the Cube of a Fraction

Calculate (2/9)^7: Evaluating Powers of Fractions

Calculate (5/8)^9: Evaluating a Fraction Raised to the Ninth Power

What's the difference between $\frac{3}{7^6}$ and $\frac{3^6}{7^6}$ ?

What if the exponent was negative, like $\left(\frac{3}{7}\right)^{-6}$ ?