Evaluate (5×6×8)^(-9): Negative Exponent Expression Problem

Q: Why can't I just make each factor negative in the exponent?

The negative exponent applies to the entire product $ (5 \times 6 \times 8) $, not each individual number. Think of it as $ (240)^{-9} $ first, then use the reciprocal rule.

Q: What's the difference between the power rule and negative exponent rule?

Use the negative exponent rule first: flip to $ \frac{1}{(5 \times 6 \times 8)^9} $. Then use the power of product rule to expand the denominator.

Q: Should I calculate 5×6×8 = 240 first?

You could get $ \frac{1}{240^9} $, but the answer $ \frac{1}{5^9 \times 6^9 \times 8^9} $ is more useful because it shows the factored form.

Q: How do I remember which rule to apply first?

Follow this order: 1) Negative exponent rule (flip to reciprocal), 2) Power of product rule (distribute the exponent). This prevents confusion with double negatives!

Negative Exponents with Product Expressions

Insert the corresponding expression:

$\left(5\times6\times8\right)^{-9}=$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:08 Let's simplify this problem together.

00:11 Remember, with negative exponents, we use the reciprocal for positive exponents.

00:16 We'll apply this to our exercise now.

00:20 First, write the reciprocal, which is 1 divided by the number.

00:24 Next, raise it to the positive exponent.

00:28 To handle an exponent over a product, let's open the parentheses.

00:32 Raise each factor inside to the given exponent.

00:36 Watch as each factor is raised to the power.

00:41 Are you following along? Great job!

00:48 And that's how we solve it!

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Insert the corresponding expression:

$\left(5\times6\times8\right)^{-9}=$

Step-by-step solution

Let's solve the problem step-by-step to find the value of $(5 \times 6 \times 8)^{-9}$ .

Step 1: Recognize that the expression $(5 \times 6 \times 8)^{-9}$ has a negative exponent. According to the negative exponent rule $a^{-b} = \frac{1}{a^b}$ , we can write:

$(5 \times 6 \times 8)^{-9} = \frac{1}{(5 \times 6 \times 8)^9}$

Step 2: Next, apply the power of a product rule. This rule states that $(xyz)^n = x^n \times y^n \times z^n$ . Therefore, apply this to $(5 \times 6 \times 8)^9$ :

$(5 \times 6 \times 8)^9 = 5^9 \times 6^9 \times 8^9$

Step 3: Substitute back into the fraction obtained in Step 1:

$\frac{1}{(5 \times 6 \times 8)^9} = \frac{1}{5^9 \times 6^9 \times 8^9}$

This result is the fully simplified expression sought for the original problem.

Therefore, the expression $(5 \times 6 \times 8)^{-9}$ simplifies to $\frac{1}{5^9 \times 6^9 \times 8^9}$ .

Final Answer

$\frac{1}{5^9\times6^9\times8^9}$

Key Points to Remember

Essential concepts to master this topic

Rule: Negative exponent means reciprocal: $a^{-n} = \frac{1}{a^n}$
Technique: Apply power of product rule: $(5 \times 6 \times 8)^9 = 5^9 \times 6^9 \times 8^9$
Check: Final answer has positive exponents in denominator: $\frac{1}{5^9 \times 6^9 \times 8^9}$ ✓

Common Mistakes

Avoid these frequent errors

Applying negative exponent to individual factors incorrectly
Don't write $(5 \times 6 \times 8)^{-9}$ as $\frac{1}{5^{-9} \times 6^{-9} \times 8^{-9}}$ = double negative! This creates negative exponents in the denominator, making the expression more complex. Always apply the negative exponent rule first: flip to reciprocal with positive exponent.

Practice Quiz

Test your knowledge with interactive questions

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

FAQ

Everything you need to know about this question

Why can't I just make each factor negative in the exponent?

The negative exponent applies to the entire product $(5 \times 6 \times 8)$ , not each individual number. Think of it as $(240)^{-9}$ first, then use the reciprocal rule.

What's the difference between the power rule and negative exponent rule?

Use the negative exponent rule first: flip to $\frac{1}{(5 \times 6 \times 8)^9}$ . Then use the power of product rule to expand the denominator.

Should I calculate 5×6×8 = 240 first?

You could get $\frac{1}{240^9}$ , but the answer $\frac{1}{5^9 \times 6^9 \times 8^9}$ is more useful because it shows the factored form.

How do I remember which rule to apply first?

Follow this order: 1) Negative exponent rule (flip to reciprocal), 2) Power of product rule (distribute the exponent). This prevents confusion with double negatives!

Why isn't the answer negative since the exponent is negative?

Negative exponents don't make the result negative! They just mean "take the reciprocal." The base $(5 \times 6 \times 8)$ is positive, so the result stays positive.

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