Solve (12×7×9)^(-5): Negative Exponent Calculation

Q: Why can't I just put the whole expression in the denominator?

Because $ \frac{1}{(12 \times 7 \times 9)^{-5}} $ actually equals $ (12 \times 7 \times 9)^{5} $! The negative exponent in the denominator becomes positive, which is the opposite of what we want.

Q: Do I need to calculate the actual numbers like 12×7×9?

No! Keep the expression factored as $ \frac{1}{12^5 \times 7^5 \times 9^5} $. This shows your understanding of the exponent rules more clearly than calculating $ 756^5 $.

Q: Why does the negative exponent make a fraction?

Negative exponents mean "take the reciprocal". So $ x^{-3} $ means "1 divided by $ x^3 $". It's like flipping the base to the opposite side of a fraction line!

Q: What if one of the factors was already a fraction?

The same rules apply! If you had $ \left(\frac{1}{2} \times 3 \times 4\right)^{-2} $, distribute the -2, then $ \left(\frac{1}{2}\right)^{-2} = 2^2 = 4 $.

Negative Exponents with Product Rule

Insert the corresponding expression:

$\left(12\times7\times9\right)^{-5}=$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Simplify the following problem

00:04 According to the laws of exponents, when we have a negative exponent

00:07 We can convert to the reciprocal number to obtain a positive exponent

00:11 We will apply this formula to our exercise

00:14 We'll write the reciprocal number (1 divided by the number)

00:17 Raise to the positive exponent

00:24 In order to expand parentheses of an exponent over multiplication

00:29 Raise each factor to the power

00:35 We will apply this formula to our exercise

00:42 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:

$\left(12\times7\times9\right)^{-5}=$

Step-by-step solution

Start with $\left(12 \times 7 \times 9\right)^{-5}$ .

Apply the power of a product property, which states $(xyz)^n = x^n \times y^n \times z^n$ :
$\left(12 \times 7 \times 9\right)^{-5} = 12^{-5} \times 7^{-5} \times 9^{-5}$

Use the negative exponent property, $a^{-n} = \frac{1}{a^n}$ :
$12^{-5} \times 7^{-5} \times 9^{-5} = \frac{1}{12^{5}} \times \frac{1}{7^{5}} \times \frac{1}{9^{5}}$

This results in:
$\frac{1}{12^5 \times 7^5 \times 9^5}$

Thus, the solution to the expression is $\frac{1}{12^5 \times 7^5 \times 9^5}$ .

Keep in mind - we could have used the rules in the other way around, first the negative exponent rule, and only then the product rule and the result would still be the same!

Final Answer

$\frac{1}{12^5\times7^5\times9^5}$

Key Points to Remember

Essential concepts to master this topic

Negative Exponent Rule: $a^{-n} = \frac{1}{a^n}$ converts negative powers to positive
Product Power Property: $(abc)^{-5} = a^{-5} \times b^{-5} \times c^{-5}$ distributes the exponent
Check: Final answer should have positive exponents in denominator: $\frac{1}{12^5 \times 7^5 \times 9^5}$ ✓

Common Mistakes

Avoid these frequent errors

Forgetting to apply the power to all factors
Don't write $\frac{1}{(12 \times 7 \times 9)^{-5}}$ = double negative gives wrong result! This creates a positive exponent in the denominator instead of the required negative. Always distribute the negative exponent to each factor first, then apply the negative exponent rule.

Practice Quiz

Test your knowledge with interactive questions

$112^0=\text{?}$

FAQ

Everything you need to know about this question

Why can't I just put the whole expression in the denominator?

Because $\frac{1}{(12 \times 7 \times 9)^{-5}}$ actually equals $(12 \times 7 \times 9)^{5}$ ! The negative exponent in the denominator becomes positive, which is the opposite of what we want.

Do I need to calculate the actual numbers like 12×7×9?

No! Keep the expression factored as $\frac{1}{12^5 \times 7^5 \times 9^5}$ . This shows your understanding of the exponent rules more clearly than calculating $756^5$ .

Which rule should I apply first - negative exponent or product rule?

Either order works! You can distribute the -5 first, then apply $a^{-n} = \frac{1}{a^n}$ , OR flip to $\frac{1}{(12 \times 7 \times 9)^5}$ first, then distribute the 5.

Why does the negative exponent make a fraction?

Negative exponents mean "take the reciprocal". So $x^{-3}$ means "1 divided by $x^3$ ". It's like flipping the base to the opposite side of a fraction line!

What if one of the factors was already a fraction?

The same rules apply! If you had $\left(\frac{1}{2} \times 3 \times 4\right)^{-2}$ , distribute the -2, then $\left(\frac{1}{2}\right)^{-2} = 2^2 = 4$ .

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