Simplify the Fraction: 1/(3⁴×12⁴) Expression Challenge

Q: Why does 1/a^n equal a^(-n) and not -a^n?

The negative exponent means 'take the reciprocal', not 'make it negative'. So $ \frac{1}{3^4} = 3^{-4} $, which is still positive but represents one divided by 3^4.

Q: Can I combine 3^(-4) and 12^(-4) into one expression?

You could write it as $ (3 \times 12)^{-4} = 36^{-4} $, but the question asks for the form that matches the original structure. Keep them separate as $ 3^{-4} \times 12^{-4} $.

Q: How do I remember the negative exponent rule?

Think of it as 'flipping': $ \frac{1}{a^n} $ flips to become $ a^{-n} $. The number moves from denominator to numerator and the exponent becomes negative.

Q: What's the difference between 3^(-4) and -3^4?

$ 3^{-4} = \frac{1}{3^4} = \frac{1}{81} $ (positive fraction), while $ -3^4 = -81 $ (negative whole number). The negative exponent creates a fraction, not a negative number!

Q: Why isn't the answer just 3^4 × 12^4?

That would be the reciprocal of what we want! We need $ \frac{1}{3^4 \times 12^4} $, which is much smaller than 1, not the large number that 3^4 × 12^4 represents.

Negative Exponents with Fraction Conversion

Insert the corresponding expression:

$\frac{1}{3^4\times12^4}=$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Simplify the following problem

00:04 Break down the fraction into 2 smaller fractions

00:14 Apply the exponent laws in order to simplify the negative exponents

00:18 Convert to the reciprocal number and raise to the power (-1)

00:21 Apply this formula to our exercise

00:27 Raise to the power (-1)

00:37 Convert to the reciprocal number (1 divided by the number)

00:56 Positive x Negative always equals a negative

01:03 This is the solution to the question

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Insert the corresponding expression:

$\frac{1}{3^4\times12^4}=$

Step-by-step solution

To address this mathematical problem, we need to rewrite the expression $\frac{1}{3^4 \times 12^4}$ using exponent properties. Specifically, we'll use the property that tells us $\frac{1}{a^n} = a^{-n}$ .

Let's perform the necessary steps:

Step 1: Apply the formula to each component in the denominator. For $\frac{1}{3^4}$ , we have $3^{-4}$ using the rule $\frac{1}{a^n} = a^{-n}$ .
Step 2: Similarly, for $\frac{1}{12^4}$ , we apply the rule to get $12^{-4}$ .
Step 3: Combine these results using multiplication: $3^{-4} \times 12^{-4}$ .

By performing these transformations, we can confirm that the expression $\frac{1}{3^4 \times 12^4}$ is equivalent to $3^{-4} \times 12^{-4}$ .

Therefore, the correct expression is $3^{-4} \times 12^{-4}$ , which is choice 3 in the multiple-choice options provided.

Final Answer

$3^{-4}\times12^{-4}$

Key Points to Remember

Essential concepts to master this topic

Rule: Convert 1/a^n to a^(-n) using negative exponent property
Technique: Apply rule separately: 1/3^4 = 3^(-4), 1/12^4 = 12^(-4)
Check: Verify 3^(-4) × 12^(-4) = 1/(3^4 × 12^4) using exponent rules ✓

Common Mistakes

Avoid these frequent errors

Adding negative signs incorrectly
Don't write -3^(-4) × 12^(-4) = wrong answer! The negative exponent rule gives positive base with negative exponent, not a negative coefficient. Always keep the base positive: 3^(-4) × 12^(-4).

Practice Quiz

Test your knowledge with interactive questions

$112^0=\text{?}$

FAQ

Everything you need to know about this question

Why does 1/a^n equal a^(-n) and not -a^n?

The negative exponent means 'take the reciprocal', not 'make it negative'. So $\frac{1}{3^4} = 3^{-4}$ , which is still positive but represents one divided by 3^4.

Can I combine 3^(-4) and 12^(-4) into one expression?

You could write it as $(3 \times 12)^{-4} = 36^{-4}$ , but the question asks for the form that matches the original structure. Keep them separate as $3^{-4} \times 12^{-4}$ .

How do I remember the negative exponent rule?

Think of it as 'flipping': $\frac{1}{a^n}$ flips to become $a^{-n}$ . The number moves from denominator to numerator and the exponent becomes negative.

What's the difference between 3^(-4) and -3^4?

$3^{-4} = \frac{1}{3^4} = \frac{1}{81}$ (positive fraction), while $-3^4 = -81$ (negative whole number). The negative exponent creates a fraction, not a negative number!

Why isn't the answer just 3^4 × 12^4?

That would be the reciprocal of what we want! We need $\frac{1}{3^4 \times 12^4}$ , which is much smaller than 1, not the large number that 3^4 × 12^4 represents.

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