Solve (1/216)^(-4): Negative Exponent with Product Fraction

Q: Why does the negative exponent disappear after flipping the fraction?

The negative exponent rule states that $ a^{-n} = \frac{1}{a^n} $. When you flip $ \frac{1}{216} $ to get $ 216 $, the negative becomes positive!

Q: How do I handle the product in the denominator?

When you have a product like $ 4\times6\times9 $ in the denominator, each factor gets the same exponent. So $ (4\times6\times9)^4 = 4^4\times6^4\times9^4 $.

Q: What if I calculated 216 first, then applied the rule?

That works too! $ \left(\frac{1}{216}\right)^{-4} = 216^4 $. But keeping it as $ 4^4\times6^4\times9^4 $ shows your work more clearly and matches the answer choices.

Q: Can I add a negative sign to the front of the expression?

No! The negative exponent rule only flips the fraction - it doesn't add a negative sign. $ \left(\frac{1}{216}\right)^{-4} $ gives a positive result.

Q: How can I remember when to flip the fraction?

Think of negative exponents as saying "flip me!" Whenever you see a negative exponent, flip the fraction and make the exponent positive. $ \left(\frac{a}{b}\right)^{-n} = \left(\frac{b}{a}\right)^n $

Negative Exponents with Product Denominators

Insert the corresponding expression:

$\left(\frac{1}{4\times6\times9}\right)^{-4}=$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Simplify the following problem

00:04 According to the exponent laws, a fraction raised to the negative power(-N)

00:08 is equal to the reciprocal raised to the opposite power(N)

00:12 We will apply this formula to our exercise

00:15 We'll convert to the reciprocal number and raise it to the opposite power

00:21 Any fraction equal to 1 is always equal to itself

00:28 According to exponent laws, a product raised to a power (N)

00:31 equals the product of its factors each raised to the power (N)

00:34 We will apply this formula to our exercise

00:38 We'll break down each product into factors and raise them to the appropriate power

00:43 That's the solution

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Insert the corresponding expression:

$\left(\frac{1}{4\times6\times9}\right)^{-4}=$

Step-by-step solution

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

Step 1: Identify the negative exponent and use the rule $\left(\frac{a}{b}\right)^{-n} = \left(\frac{b}{a}\right)^n$ .
Step 2: Apply the rule to the expression $\left(\frac{1}{4 \times 6 \times 9}\right)^{-4}$ .
Step 3: Simplify the result.

Now, let's work through each step:
Step 1: The given expression is $\left(\frac{1}{4 \times 6 \times 9}\right)^{-4}$ . Notice the negative exponent $-4$ .
Step 2: According to the rule, flipping the fraction and changing the sign of the exponent, we get $\left(4 \times 6 \times 9\right)^{4}$ .
Step 3: Thus, the expression simplifies to $4^4 \times 6^4 \times 9^4$ .

Therefore, the solution to the problem is $4^4 \times 6^4 \times 9^4$ , which corresponds to choice 2.

Final Answer

$4^4\times6^4\times9^4$

Key Points to Remember

Essential concepts to master this topic

Rule: Negative exponent flips fraction and makes exponent positive
Technique: Transform $\left(\frac{1}{4\times6\times9}\right)^{-4}$ to $(4\times6\times9)^4$
Check: Product in denominator becomes separate positive exponents: $4^4\times6^4\times9^4$ ✓

Common Mistakes

Avoid these frequent errors

Keeping the negative sign in the exponents
Don't write $4^{-4}\times6^{-4}\times9^{-4}$ after flipping the fraction = wrong answer! The negative exponent disappears when you flip the fraction. Always make the exponent positive after applying the negative exponent rule.

Practice Quiz

Test your knowledge with interactive questions

$112^0=\text{?}$

FAQ

Everything you need to know about this question

Why does the negative exponent disappear after flipping the fraction?

The negative exponent rule states that $a^{-n} = \frac{1}{a^n}$ . When you flip $\frac{1}{216}$ to get $216$ , the negative becomes positive!

How do I handle the product in the denominator?

When you have a product like $4\times6\times9$ in the denominator, each factor gets the same exponent. So $(4\times6\times9)^4 = 4^4\times6^4\times9^4$ .

What if I calculated 216 first, then applied the rule?

That works too! $\left(\frac{1}{216}\right)^{-4} = 216^4$ . But keeping it as $4^4\times6^4\times9^4$ shows your work more clearly and matches the answer choices.

Can I add a negative sign to the front of the expression?

No! The negative exponent rule only flips the fraction - it doesn't add a negative sign. $\left(\frac{1}{216}\right)^{-4}$ gives a positive result.

How can I remember when to flip the fraction?

Think of negative exponents as saying "flip me!" Whenever you see a negative exponent, flip the fraction and make the exponent positive. $\left(\frac{a}{b}\right)^{-n} = \left(\frac{b}{a}\right)^n$

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