Solve (1/24)^(-2): Negative Exponent with Product Denominator

Q: Why doesn't the negative exponent make the answer negative?

The negative exponent doesn't change the sign - it just means "flip the fraction"! Think of it as moving from bottom to top of a fraction, not making numbers negative.

Q: Do I need to calculate the actual numbers like 4×9×16?

Usually no! Leave your answer as $ 2^2\times3^2\times4^2 $ unless specifically asked to simplify further. This form clearly shows your work.

Q: What if the denominator had different numbers?

The same rule applies! For any fraction $ \frac{1}{abc} $ with negative exponent, you get $ (abc)^{positive\,exponent} $.

Q: How do I remember which way to flip?

Think: negative exponent = opposite position. If it starts on bottom (denominator), negative exponent moves it to top (numerator). Easy!

Q: What happens to the exponent when I flip the fraction?

The exponent becomes positive! Flipping the fraction "cancels out" the negative, so $ -2 $ becomes $ +2 $.

Negative Exponents with Fraction Reciprocals

Insert the corresponding expression:

$\left(\frac{1}{2\times3\times4}\right)^{-2}=$

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Step-by-step written solution

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Understand the problem

Insert the corresponding expression:

$\left(\frac{1}{2\times3\times4}\right)^{-2}=$

Step-by-step solution

We are given the expression: $\left(\frac{1}{2\times3\times4}\right)^{-2}$ . We need to simplify it using the rules of exponents.

Step 1: Identify the base of the exponent.
The base is $\frac{1}{2\times3\times4}$ .
Step 2: Apply the rule for negative exponents.
For a fraction $\frac{1}{a}$ with a negative exponent, $\left( \frac{1}{a} \right)^{-n} = a^n$ . Therefore, $\left(\frac{1}{2\times3\times4}\right)^{-2} = (2\times3\times4)^2$
Step 3: Expand the expression.
$(2\times3\times4)^2 = 2^2 \times 3^2 \times 4^2$ .

Thus, the simplified expression is: $2^2\times3^2\times4^2$

Final Answer

$2^2\times3^2\times4^2$

Key Points to Remember

Essential concepts to master this topic

Rule: Negative exponent on fraction flips base and makes exponent positive
Technique: $\left(\frac{1}{a}\right)^{-n} = a^n$ , so flip fraction and remove negative
Check: Substitute back: $2^2\times3^2\times4^2 = 576$ equals $\frac{1}{576^{-1}}$ ✓

Common Mistakes

Avoid these frequent errors

Adding negative sign to final answer
Don't write - $(2\times3\times4)^2$ = negative result! The negative exponent doesn't make the answer negative - it just means "take reciprocal". Always remember that negative exponents only affect position (numerator vs denominator), not sign.

Practice Quiz

Test your knowledge with interactive questions

$112^0=\text{?}$

FAQ

Everything you need to know about this question

Why doesn't the negative exponent make the answer negative?

The negative exponent doesn't change the sign - it just means "flip the fraction"! Think of it as moving from bottom to top of a fraction, not making numbers negative.

Do I need to calculate the actual numbers like 4×9×16?

Usually no! Leave your answer as $2^2\times3^2\times4^2$ unless specifically asked to simplify further. This form clearly shows your work.

What if the denominator had different numbers?

The same rule applies! For any fraction $\frac{1}{abc}$ with negative exponent, you get $(abc)^{positive\,exponent}$ .

How do I remember which way to flip?

Think: negative exponent = opposite position. If it starts on bottom (denominator), negative exponent moves it to top (numerator). Easy!

What happens to the exponent when I flip the fraction?

The exponent becomes positive! Flipping the fraction "cancels out" the negative, so $-2$ becomes $+2$ .

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