Solve (9×7×8)^(-8): Negative Exponent Expression Challenge

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

The negative exponent doesn't change the sign of the result! It just means "take the reciprocal." So $ 9^{-8} $ equals $ \frac{1}{9^8} $, which is still positive.

Q: Do I distribute the -8 to each number inside the parentheses?

Yes! When you have $ (a \times b \times c)^n $, the exponent n applies to each factor separately. So $ (9 \times 7 \times 8)^{-8} = 9^{-8} \times 7^{-8} \times 8^{-8} $.

Q: What if I calculated the product first, then applied the exponent?

You could calculate $ 9 \times 7 \times 8 = 504 $ first, then write $ 504^{-8} $. Both methods are correct, but distributing the exponent often makes the problem easier to work with!

Q: How do I know when to use the power of a product rule?

Use this rule whenever you see multiplication inside parentheses with an exponent outside. The pattern is $ (factor_1 \times factor_2 \times ...)^{exponent} $.

Q: What's the difference between this and adding exponents?

Adding exponents is for same bases: $ x^2 \times x^3 = x^5 $. Here we have different bases (9, 7, 8) with the same exponent, so we use the power of a product rule instead.

Power of Product with Negative Exponents

Insert the corresponding expression:

$\left(9\times7\times8\right)^{-8}=$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:09 Let's simplify this problem together.

00:12 To deal with parentheses and a multiplying exponent outside,

00:17 we raise each part inside to that power.

00:20 Let's apply this step-by-step to our exercise.

00:31 And that's how we find the solution.

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Insert the corresponding expression:

$\left(9\times7\times8\right)^{-8}=$

Step-by-step solution

To solve this problem, we will apply the rule for the power of a product and handle the negative exponent:

The expression given is $(9 \times 7 \times 8)^{-8}$ .
According to the Power of a Product rule: $(a \times b \times c)^n = a^n \times b^n \times c^n$ where $a$ , $b$ , and $c$ are the factors inside the parentheses, and $n$ is the exponent.
Applying this rule, we distribute the exponent $-8$ to each of the factors inside the parentheses:
$(9 \times 7 \times 8)^{-8} = 9^{-8} \times 7^{-8} \times 8^{-8}$ .

Therefore, the solution to the problem is $9^{-8} \times 7^{-8} \times 8^{-8}$ .

Final Answer

$9^{-8}\times7^{-8}\times8^{-8}$

Key Points to Remember

Essential concepts to master this topic

Power of Product Rule: $(a \times b \times c)^n = a^n \times b^n \times c^n$
Technique: Distribute the exponent -8 to each factor: $9^{-8} \times 7^{-8} \times 8^{-8}$
Check: Each factor gets the same exponent as the original parentheses ✓

Common Mistakes

Avoid these frequent errors

Changing the sign when distributing negative exponents
Don't write $-9^{-8} \times 7^{-8} \times 8^{-8}$ or $-9^8 \times 7^8 \times 8^8$ = wrong answer! The negative is part of the exponent, not a negative sign for the base. Always keep the exponent exactly as written when distributing.

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 of the result! It just means "take the reciprocal." So $9^{-8}$ equals $\frac{1}{9^8}$ , which is still positive.

Do I distribute the -8 to each number inside the parentheses?

Yes! When you have $(a \times b \times c)^n$ , the exponent n applies to each factor separately. So $(9 \times 7 \times 8)^{-8} = 9^{-8} \times 7^{-8} \times 8^{-8}$ .

What if I calculated the product first, then applied the exponent?

You could calculate $9 \times 7 \times 8 = 504$ first, then write $504^{-8}$ . Both methods are correct, but distributing the exponent often makes the problem easier to work with!

How do I know when to use the power of a product rule?

Use this rule whenever you see multiplication inside parentheses with an exponent outside. The pattern is $(factor_1 \times factor_2 \times ...)^{exponent}$ .

What's the difference between this and adding exponents?

Adding exponents is for same bases: $x^2 \times x^3 = x^5$ . Here we have different bases (9, 7, 8) with the same exponent, so we use the power of a product rule instead.

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