Simplify the Power Fraction: (20^4)/(31^4) Calculation

Q: How do I know when to use the quotient rule for exponents?

Use it when you have the same exponent on both the numerator and denominator! If you see $ \frac{a^n}{b^n} $, it equals $ \left(\frac{a}{b}\right)^n $.

Q: What if the exponents in the numerator and denominator are different?

Then you cannot use this rule! For example, $ \frac{20^4}{31^3} $ stays as is and doesn't simplify to $ \left(\frac{20}{31}\right)^{anything} $.

Q: Can I work backwards to check my answer?

Absolutely! Expand $ \left(\frac{20}{31}\right)^4 $ by multiplying: $ \frac{20}{31} \times \frac{20}{31} \times \frac{20}{31} \times \frac{20}{31} = \frac{20^4}{31^4} $ ✓

Q: Why can't I just cancel the 4s in the exponents?

Because exponents aren't fractions to reduce! The 4s tell you how many times to multiply the base. You're looking for a way to rewrite the expression, not simplify by canceling.

Q: Does this work with any numbers and exponents?

Yes! The rule $ \frac{a^n}{b^n} = \left(\frac{a}{b}\right)^n $ works for any real numbers a, b (where b ≠ 0) and any positive integer n.

Fraction Exponents with Quotient Rule

Insert the corresponding expression:

$\frac{20^4}{31^4}=$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:08 Let's simplify this problem together.

00:13 Remember, with exponent laws, if a fraction is raised to the power of N

00:19 it means both the numerator and the denominator are raised to that same power N.

00:26 We'll use this idea now, but in reverse, to solve the problem.

00:33 And here is our solution. Great job!

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Insert the corresponding expression:

$\frac{20^4}{31^4}=$

Step-by-step solution

To solve this problem, we need to rewrite the given expression $\frac{20^4}{31^4}$ using properties of exponents.

Let's take these steps:

Step 1: Recognize the expression as a fraction raised to a power. The problem provides $\frac{20^4}{31^4}$ .
Step 2: Apply the power of a fraction rule: For any real numbers $a$ and $b$ , and a positive integer $n$ , $\left( \frac{a}{b} \right)^n = \frac{a^n}{b^n}$ .

Applying Step 2, we write:

$\frac{20^4}{31^4} = \left(\frac{20}{31}\right)^4$ .

Thus, the corresponding expression is $\left(\frac{20}{31}\right)^4$ .

Therefore, the solution to the problem is $\left(\frac{20}{31}\right)^4$ .

Final Answer

$\left(\frac{20}{31}\right)^4$

Key Points to Remember

Essential concepts to master this topic

Quotient Rule: $\frac{a^n}{b^n} = \left(\frac{a}{b}\right)^n$ when exponents are equal
Technique: Recognize $\frac{20^4}{31^4}$ as $\left(\frac{20}{31}\right)^4$
Check: Expand $\left(\frac{20}{31}\right)^4 = \frac{20^4}{31^4}$ ✓

Common Mistakes

Avoid these frequent errors

Incorrectly distributing exponents to separate terms
Don't write $\frac{20^4}{31^4}$ as $4\times\left(\frac{20}{31}\right)^3$ or $\frac{20\times4}{31^4}$ = wrong expression! These change the mathematical meaning completely. Always recognize that when both numerator and denominator have the same exponent, you can factor it out as the power of the entire fraction.

Practice Quiz

Test your knowledge with interactive questions

$112^0=\text{?}$

FAQ

Everything you need to know about this question

How do I know when to use the quotient rule for exponents?

Use it when you have the same exponent on both the numerator and denominator! If you see $\frac{a^n}{b^n}$ , it equals $\left(\frac{a}{b}\right)^n$ .

What if the exponents in the numerator and denominator are different?

Then you cannot use this rule! For example, $\frac{20^4}{31^3}$ stays as is and doesn't simplify to $\left(\frac{20}{31}\right)^{anything}$ .

Can I work backwards to check my answer?

Absolutely! Expand $\left(\frac{20}{31}\right)^4$ by multiplying: $\frac{20}{31} \times \frac{20}{31} \times \frac{20}{31} \times \frac{20}{31} = \frac{20^4}{31^4}$ ✓

Why can't I just cancel the 4s in the exponents?

Because exponents aren't fractions to reduce! The 4s tell you how many times to multiply the base. You're looking for a way to rewrite the expression, not simplify by canceling.

Does this work with any numbers and exponents?

Yes! The rule $\frac{a^n}{b^n} = \left(\frac{a}{b}\right)^n$ works for any real numbers a, b (where b ≠ 0) and any positive integer n.

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