Evaluate the Expression: (1/6)^5 Fraction Calculation

Fraction Powers with Exponent Rules

Insert the corresponding expression:

1565= \frac{1^5}{6^5}=

❤️ 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
00:00 Simplify the following problem
00:03 According to the laws of exponents, a fraction raised to a power (N)
00:07 equals the numerator and denominator, raised to the same power (N)
00:14 We'll apply this formula to our exercise, only this time in the opposite direction
00:18 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:

1565= \frac{1^5}{6^5}=

2

Step-by-step solution

To solve this problem, we need to express 1565 \frac{1^5}{6^5} using the power of a fraction rule:

  • Step 1: Identify that both the numerator and denominator are raised to the same power, 5.
  • Step 2: Recognize that the expression can be rewritten as (16)5 \left(\frac{1}{6}\right)^5 using the power of a fraction rule: (ab)n=anbn\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}.

Applying the formula, we convert 1565 \frac{1^5}{6^5} into (16)5 \left(\frac{1}{6}\right)^5 .

Therefore, the solution to the problem and correct multiple-choice answer is (16)5 \left(\frac{1}{6}\right)^5 , which corresponds to choice 2.

3

Final Answer

(16)5 \left(\frac{1}{6}\right)^5

Key Points to Remember

Essential concepts to master this topic
  • Rule: Power of fraction equals fraction raised to that power
  • Technique: anbn=(ab)n \frac{a^n}{b^n} = \left(\frac{a}{b}\right)^n converts separate powers to single power
  • Check: Both forms give same result: 17776 \frac{1}{7776} ✓

Common Mistakes

Avoid these frequent errors
  • Combining exponents incorrectly with denominators
    Don't write 156×5 \frac{1^5}{6×5} or 156×65 \frac{1^5}{6×6^5} = wrong operations! This confuses multiplication with exponentiation rules. Always recognize that anbn \frac{a^n}{b^n} becomes (ab)n \left(\frac{a}{b}\right)^n .

Practice Quiz

Test your knowledge with interactive questions

\( 112^0=\text{?} \)

FAQ

Everything you need to know about this question

Why can I write 1565 \frac{1^5}{6^5} as (16)5 \left(\frac{1}{6}\right)^5 ?

+

This uses the power of a quotient rule! When both numerator and denominator have the same exponent, you can combine them: anbn=(ab)n \frac{a^n}{b^n} = \left(\frac{a}{b}\right)^n .

What's the actual value of this expression?

+

Both forms equal 17776 \frac{1}{7776} . Since 15=1 1^5 = 1 and 65=7776 6^5 = 7776 , we get 17776 \frac{1}{7776} .

Does this rule work with any exponent?

+

Yes! The rule anbn=(ab)n \frac{a^n}{b^n} = \left(\frac{a}{b}\right)^n works for any exponent - positive, negative, or even fractional exponents.

Which form is better to use?

+

It depends on the problem! (16)5 \left(\frac{1}{6}\right)^5 is more compact, while 1565 \frac{1^5}{6^5} shows the individual operations more clearly.

Can I use this rule backwards too?

+

Absolutely! You can go from (34)2 \left(\frac{3}{4}\right)^2 to 3242=916 \frac{3^2}{4^2} = \frac{9}{16} . Both directions are useful in different situations.

🌟 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

More Questions

Click on any question to see the complete solution with step-by-step explanations

Powers of a Fraction

Simplify the Power Fraction: (10^5)/(17^5) Expression
Click to solve
Calculate (3/80)^5: Solving a Compound Fraction Power Expression
Click to solve