Solve (3/280)^(-7): Negative Exponent with Complex Fraction

Q: Why do I need to flip the fraction when there's a negative exponent?

A negative exponent means "take the reciprocal." So $ (\frac{3}{280})^{-7} $ literally means "the reciprocal of this fraction, raised to the 7th power."

Q: What does 5×8×7 equal, and do I need to calculate it?

No calculation needed! Since 5×8×7 = 280, you can leave it as 5×8×7 or write 280. The problem asks for the expression form, not a numerical answer.

Q: How do I remember which way to flip the fraction?

Think of it as "negative means opposite." The opposite of $ \frac{3}{280} $ is $ \frac{280}{3} $. Then change the -7 to +7!

Q: Can I distribute the exponent to each number separately?

Yes! $ (\frac{5×8×7}{3})^7 $ equals $ \frac{5^7×8^7×7^7}{3^7} $, but the first form is usually preferred because it's simpler to read.

Q: What if I accidentally wrote the reciprocal wrong?

Double-check by asking: "Did the top and bottom switch places?" Original: $ \frac{3}{5×8×7} $. Flipped: $ \frac{5×8×7}{3} $. ✓

Negative Exponents with Reciprocal Rule

Insert the corresponding expression:

$\left(\frac{3}{5\times8\times7}\right)^{-7}=$

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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 exponent laws, a fraction raised to the negative power (-N)

00:07 is equal to the reciprocal fraction raised to the opposite exponent (N)

00:11 We will apply this formula to our exercise

00:14 We will convert to a reciprocal number and raise it to the opposite exponent

00:20 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:

$\left(\frac{3}{5\times8\times7}\right)^{-7}=$

Step-by-step solution

The goal is to simplify the expression $\left(\frac{3}{5 \times 8 \times 7}\right)^{-7}$ by converting the negative exponent to a positive one.

Step 1: Apply the negative exponent rule:
The expression $\left(\frac{3}{5 \times 8 \times 7}\right)^{-7}$ implies we take the reciprocal of the fraction and change the exponent to positive:
$\left(\frac{3}{5 \times 8 \times 7}\right)^{-7} = \left(\frac{5 \times 8 \times 7}{3}\right)^7$ .

Step 2: Simplify the expression:
By applying the negative exponent rule correctly, the expression is simplified correctly to $\left(\frac{5 \times 8 \times 7}{3}\right)^7$ .

Thus, the expression $\left(\frac{3}{5 \times 8 \times 7}\right)^{-7}$ simplifies to:

$\left(\frac{5 \times 8 \times 7}{3}\right)^7$ .

Final Answer

$\left(\frac{5\times8\times7}{3}\right)^7$

Key Points to Remember

Essential concepts to master this topic

Rule: Negative exponent means take the reciprocal and make positive
Technique: $(\frac{a}{b})^{-n} = (\frac{b}{a})^n$ - flip numerator and denominator
Check: Result has positive exponent and reciprocal base structure ✓

Common Mistakes

Avoid these frequent errors

Keeping fraction the same and just removing negative sign
Don't write $(\frac{3}{280})^{-7} = (\frac{3}{280})^7$ = wrong direction! This ignores the reciprocal rule and gives the opposite of the correct answer. Always flip the fraction first, then make the exponent positive.

Practice Quiz

Test your knowledge with interactive questions

$112^0=\text{?}$

FAQ

Everything you need to know about this question

Why do I need to flip the fraction when there's a negative exponent?

A negative exponent means "take the reciprocal." So $(\frac{3}{280})^{-7}$ literally means "the reciprocal of this fraction, raised to the 7th power."

What does 5×8×7 equal, and do I need to calculate it?

No calculation needed! Since 5×8×7 = 280, you can leave it as 5×8×7 or write 280. The problem asks for the expression form, not a numerical answer.

How do I remember which way to flip the fraction?

Think of it as "negative means opposite." The opposite of $\frac{3}{280}$ is $\frac{280}{3}$ . Then change the -7 to +7!

Can I distribute the exponent to each number separately?

Yes! $(\frac{5×8×7}{3})^7$ equals $\frac{5^7×8^7×7^7}{3^7}$ , but the first form is usually preferred because it's simpler to read.

What if I accidentally wrote the reciprocal wrong?

Double-check by asking: "Did the top and bottom switch places?" Original: $\frac{3}{5×8×7}$ . Flipped: $\frac{5×8×7}{3}$ . ✓

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