Simplify 1/(8⁻⁷ × 9⁻⁷ × 5⁻⁷): Negative Exponent Practice

Q: Why does dividing by a negative exponent make it positive?

Think of it as double negatives canceling out! When you have $ \frac{1}{a^{-n}} $, you're dividing by something that's already "flipped," so you flip it back to get $ a^n $.

Q: Can I combine the numbers first, then apply the exponent?

Yes! Since all terms have the same exponent (-7), you can use the power of a product rule: $ a^n \times b^n \times c^n = (abc)^n $. This makes calculations much easier!

Q: How do I know when to use the reciprocal rule?

Use the reciprocal rule whenever you see negative exponents in a denominator. The pattern $ \frac{1}{a^{-n}} = a^n $ always works and simplifies your work significantly!

Q: Should I calculate 8 × 9 × 5 first?

Not necessary! The question asks for the expression, not the final number. Leaving it as $ (8 \times 9 \times 5)^7 $ shows you understand the concept and is the correct simplified form.

Negative Exponent Rules with Multiple Terms

Insert the corresponding expression:

$\frac{1}{8^{-7}\times9^{-7}\times5^{-7}}=$

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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 A product where each factor is raised to the same power (N)

00:08 Can be converted to parentheses of the entire product raised to the power of the factor (N)

00:14 We will apply this formula to our exercise

00:25 Apply the exponent laws in order to simplify the negative exponents

00:29 Convert to the reciprocal number and raise to the power multiplied by (-1)

00:32 We will apply this formula to our exercise

00:35 Convert to the reciprocal number (1 divided by the number)

00:41 Raise to the power multiplied by (-1)

00:44 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:

$\frac{1}{8^{-7}\times9^{-7}\times5^{-7}}=$

Step-by-step solution

To solve the problem, follow these steps:

Given the expression $\frac{1}{8^{-7} \times 9^{-7} \times 5^{-7}}$ .
Apply the negative exponent rule: Each term in the denominator is raised to a negative power.
Write each term with positive exponents using the reciprocal rule: $\frac{1}{a^{-n}} = a^n$ .
This results in the expression: $8^7 \times 9^7 \times 5^7$ .
The power of a product rule tells us that this is equivalent to $(8 \times 9 \times 5)^7$ .

Therefore, the solution is $\left(8 \times 9 \times 5\right)^7$ .

Comparing with the multiple-choice options provided, the correct choice is: $\left(8 \times 9 \times 5\right)^7$ .

Final Answer

$\left(8\times9\times5\right)^7$

Key Points to Remember

Essential concepts to master this topic

Rule: Negative exponents in denominators become positive when moved to numerator
Technique: Convert $\frac{1}{8^{-7}}$ to $8^7$ using reciprocal rule
Check: Use power of product rule: $a^n \times b^n \times c^n = (a \times b \times c)^n$ ✓

Common Mistakes

Avoid these frequent errors

Ignoring the fraction bar and treating negative exponents individually
Don't work with 8^(-7) × 9^(-7) × 5^(-7) separately = gets messy fractions! This makes the problem much harder than needed. Always apply the reciprocal rule first: 1/(a^(-n)) = a^n for each term.

Practice Quiz

Test your knowledge with interactive questions

$$ (4^2)^3+(g^3)^4= $$

FAQ

Everything you need to know about this question

Why does dividing by a negative exponent make it positive?

Think of it as double negatives canceling out! When you have $\frac{1}{a^{-n}}$ , you're dividing by something that's already "flipped," so you flip it back to get $a^n$ .

Can I combine the numbers first, then apply the exponent?

Yes! Since all terms have the same exponent (-7), you can use the power of a product rule: $a^n \times b^n \times c^n = (abc)^n$ . This makes calculations much easier!

What if the exponents were different numbers?

If the exponents were different, you'd have to handle each term separately. But when they're the same (like -7 here), you can factor out the common exponent using product rules.

How do I know when to use the reciprocal rule?

Use the reciprocal rule whenever you see negative exponents in a denominator. The pattern $\frac{1}{a^{-n}} = a^n$ always works and simplifies your work significantly!

Should I calculate 8 × 9 × 5 first?

Not necessary! The question asks for the expression, not the final number. Leaving it as $(8 \times 9 \times 5)^7$ shows you understand the concept and is the correct simplified form.

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