Simplify (b/5)⁴: Evaluating the Fourth Power of a Fraction

Exponent Rules with Fractional Bases

Insert the corresponding expression:

(b5)4= \left(\frac{b}{5}\right)^4=

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations
00:07 Let's simplify this problem together.
00:10 Remember, when a fraction is raised to a power, like power N,
00:15 we raise both the numerator and the denominator to that power.
00:19 We'll use this rule for our exercise now.
00:22 Let's raise the numerator and the denominator to power N.
00:27 And there you have it, that's the solution!

Step-by-step written solution

Follow each step carefully to understand the complete solution
1

Understand the problem

Insert the corresponding expression:

(b5)4= \left(\frac{b}{5}\right)^4=

2

Step-by-step solution

To solve this problem, we'll apply the exponent rule for fractions:

  • Step 1: Identify the fraction b5\frac{b}{5} and the power 44.
  • Step 2: Apply the exponent to both the numerator and the denominator, as per the formula.
  • Step 3: Use the rule (b5)4=b454 \left(\frac{b}{5}\right)^4 = \frac{b^4}{5^4} .

Now, let's work through the application:
Step 1: We have the base fraction b5\frac{b}{5} and exponent 44.
Step 2: According to the exponent rule, (ab)n=anbn \left(\frac{a}{b}\right)^n = \frac{a^n}{b^n} , apply the exponent 44 to both bb and 55.
Step 3: This results in the expression b454\frac{b^4}{5^4}.

Therefore, the expression (b5)4 \left(\frac{b}{5}\right)^4 simplifies to b454 \frac{b^4}{5^4} .

3

Final Answer

b454 \frac{b^4}{5^4}

Key Points to Remember

Essential concepts to master this topic
  • Rule: When raising a fraction to a power, apply the exponent to both numerator and denominator
  • Technique: Use (ab)n=anbn \left(\frac{a}{b}\right)^n = \frac{a^n}{b^n} so (b5)4=b454 \left(\frac{b}{5}\right)^4 = \frac{b^4}{5^4}
  • Check: Verify both parts have the same exponent: numerator b⁴ and denominator 5⁴ ✓

Common Mistakes

Avoid these frequent errors
  • Applying exponent to only the numerator
    Don't just raise b to the 4th power = b45 \frac{b^4}{5} ! This ignores the denominator and gives the wrong result. Always apply the exponent to both the numerator AND denominator when raising a fraction to a power.

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 of the fraction?

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Because (b5)4 \left(\frac{b}{5}\right)^4 means multiplying the entire fraction by itself 4 times! Each multiplication affects both the top and bottom, so the exponent must go on both parts.

What's the difference between b45 \frac{b^4}{5} and b454 \frac{b^4}{5^4} ?

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Huge difference! b45 \frac{b^4}{5} means only the numerator was raised to the 4th power, while b454 \frac{b^4}{5^4} correctly shows both parts raised to the 4th power. Only the second one is right!

Do I need to calculate what 5⁴ equals?

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Not necessarily! The expression b454 \frac{b^4}{5^4} is already simplified. You could calculate 54=625 5^4 = 625 to write b4625 \frac{b^4}{625} , but both forms are correct.

Does this rule work for any exponent, not just 4?

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Yes! The rule (ab)n=anbn \left(\frac{a}{b}\right)^n = \frac{a^n}{b^n} works for any exponent n, whether it's 2, 3, 10, or even negative numbers.

What if the numerator or denominator is already raised to a power?

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Use the power of a power rule! For example, (x23)4=(x2)434=x834 \left(\frac{x^2}{3}\right)^4 = \frac{(x^2)^4}{3^4} = \frac{x^8}{3^4} . Multiply the exponents: 2 × 4 = 8.

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