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

Q: Why does the exponent go to both the top and bottom of the fraction?

Because exponents distribute over division! Think of it as $\left(\frac{20}{21}\right)^4 = \frac{20}{21} \times \frac{20}{21} \times \frac{20}{21} \times \frac{20}{21}$, which gives you $\frac{20 \times 20 \times 20 \times 20}{21 \times 21 \times 21 \times 21} = \frac{20^4}{21^4}$.

Q: Do I need to calculate the actual values of 20⁴ and 21⁴?

Not unless asked! The expression $\frac{20^4}{21^4}$ is already in its simplest form for most purposes. Only calculate the numbers if the problem specifically asks for a decimal answer.

Q: What's the difference between (20/21)⁴ and 4×(20/21)?

Huge difference! $\left(\frac{20}{21}\right)^4$ means multiply the fraction by itself 4 times, while $4 \times \frac{20}{21}$ means multiply the fraction by 4. Exponents and multiplication are completely different operations!

Q: How do I remember this exponent rule?

Think: "Exponents go everywhere in fractions!" Whatever power you see outside the fraction gets applied to both the numerator and denominator. It's like the exponent can't pick favorites!

Exponent Rules with Fractional Bases

Insert the corresponding expression:

$\left(\frac{20}{21}\right)^4=$

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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 the power (N)

00:06 equals the numerator and denominator, each raised to the same power (N)

00:10 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

Understand the problem

Insert the corresponding expression:

$\left(\frac{20}{21}\right)^4=$

Step-by-step solution

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

Identify the given expression which is $\left(\frac{20}{21}\right)^4$ .
Apply the exponentiation rule for fractions: $\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$ .
Calculate $20^4$ and $21^4$ and place them as the numerator and denominator, respectively.

Now, let's work through each step:
Step 1: We begin with the expression $\left(\frac{20}{21}\right)^4$ .
Step 2: Using the power of a fraction rule, we have $\left(\frac{20}{21}\right)^4 = \frac{20^4}{21^4}$ .

Therefore, the corresponding simplified expression is $\frac{20^4}{21^4}$ .

Final Answer

$\frac{20^4}{21^4}$

Key Points to Remember

Essential concepts to master this topic

Rule: Apply exponent to both numerator and denominator separately
Technique: $\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$ means $\left(\frac{20}{21}\right)^4 = \frac{20^4}{21^4}$
Check: Verify exponent applies to both parts: top becomes $20^4$ , bottom becomes $21^4$ ✓

Common Mistakes

Avoid these frequent errors

Applying exponent to only numerator or denominator
Don't write $\left(\frac{20}{21}\right)^4 = \frac{20^4}{21}$ or $\frac{20}{21^4}$ = completely wrong answer! The exponent must distribute to both parts of the fraction. Always apply the exponent to both numerator AND denominator: $\frac{20^4}{21^4}$ .

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 does the exponent go to both the top and bottom of the fraction?

Because exponents distribute over division! Think of it as $\left(\frac{20}{21}\right)^4 = \frac{20}{21} \times \frac{20}{21} \times \frac{20}{21} \times \frac{20}{21}$ , which gives you $\frac{20 \times 20 \times 20 \times 20}{21 \times 21 \times 21 \times 21} = \frac{20^4}{21^4}$ .

Do I need to calculate the actual values of 20⁴ and 21⁴?

Not unless asked! The expression $\frac{20^4}{21^4}$ is already in its simplest form for most purposes. Only calculate the numbers if the problem specifically asks for a decimal answer.

Can I simplify this fraction before applying the exponent?

You could, but it won't help much here! Since 20 and 21 don't share common factors (20 = 2² × 5, 21 = 3 × 7), the fraction $\frac{20}{21}$ is already in lowest terms.

What's the difference between (20/21)⁴ and 4×(20/21)?

Huge difference! $\left(\frac{20}{21}\right)^4$ means multiply the fraction by itself 4 times, while $4 \times \frac{20}{21}$ means multiply the fraction by 4. Exponents and multiplication are completely different operations!

How do I remember this exponent rule?

Think: "Exponents go everywhere in fractions!" Whatever power you see outside the fraction gets applied to both the numerator and denominator. It's like the exponent can't pick favorites!

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