Solve (10×7×9)^(-5): Negative Exponent Expression Challenge

Q: Does the negative exponent make the answer negative?

No! A negative exponent creates a reciprocal, not a negative number. $ a^{-n} $ means $ \frac{1}{a^n} $, which is always positive when the base is positive.

Q: What happens to the multiplication inside the parentheses?

The entire product $ (10×7×9) $ stays together as one base. Don't separate the factors! The negative exponent applies to the whole grouped expression.

Q: Can I simplify 10×7×9 first?

You could calculate $ 10×7×9 = 630 $ to get $ 630^{-5} = \frac{1}{630^5} $, but it's usually better to leave it factored for easier reading.

Q: Is this the same as dividing by the base raised to 5?

Exactly! $ (10×7×9)^{-5} $ means dividing 1 by $ (10×7×9)^5 $. Think of it as "one over the base to the positive power."

Q: What if I see a double negative like in answer choice 4?

Be careful! $ \frac{1}{(10×7×9)^{-5}} $ would equal $ (10×7×9)^5 $ because you're taking the reciprocal of a reciprocal, which brings you back to the original positive power.

Negative Exponents with Multiple Base Factors

Insert the corresponding expression:

$\left(10\times7\times9\right)^{-5}=$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Simplify the following problem

00:03 According to the exponent laws, when we have a negative exponent

00:06 We can convert to the reciprocal number and obtain a positive exponent

00:09 We will apply this formula to our exercise

00:12 We'll write the reciprocal number (1 divided by the number)

00:17 Proceed to raise it to the positive exponent

00:23 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:

$\left(10\times7\times9\right)^{-5}=$

Step-by-step solution

We have the expression $\left(10 \times 7 \times 9\right)^{-5}$ .

According to the rule for negative exponents, which states that $a^{-n} = \frac{1}{a^n}$ , this expression can be rewritten as the reciprocal:

$\left(10 \times 7 \times 9\right)^{-5} = \frac{1}{\left(10 \times 7 \times 9\right)^5}$ .

Thus, the simplified expression is:

$\frac{1}{\left(10 \times 7 \times 9\right)^5}$ .

Final Answer

$\frac{1}{\left(10\times7\times9\right)^5}$

Key Points to Remember

Essential concepts to master this topic

Rule: Negative exponent means reciprocal: $a^{-n} = \frac{1}{a^n}$
Technique: Keep base unchanged, flip to denominator with positive exponent
Check: Verify the base stays grouped: $(10×7×9)^{-5} = \frac{1}{(10×7×9)^5}$ ✓

Common Mistakes

Avoid these frequent errors

Making the entire expression negative
Don't think $(10×7×9)^{-5} = -(10×7×9)^5$ ! The negative exponent doesn't make the result negative, it creates a reciprocal. Always remember: negative exponent = flip to denominator 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

Does the negative exponent make the answer negative?

No! A negative exponent creates a reciprocal, not a negative number. $a^{-n}$ means $\frac{1}{a^n}$ , which is always positive when the base is positive.

What happens to the multiplication inside the parentheses?

The entire product $(10×7×9)$ stays together as one base. Don't separate the factors! The negative exponent applies to the whole grouped expression.

Can I simplify 10×7×9 first?

You could calculate $10×7×9 = 630$ to get $630^{-5} = \frac{1}{630^5}$ , but it's usually better to leave it factored for easier reading.

Is this the same as dividing by the base raised to 5?

Exactly! $(10×7×9)^{-5}$ means dividing 1 by $(10×7×9)^5$ . Think of it as "one over the base to the positive power."

What if I see a double negative like in answer choice 4?

Be careful! $\frac{1}{(10×7×9)^{-5}}$ would equal $(10×7×9)^5$ because you're taking the reciprocal of a reciprocal, which brings you back to the original positive power.

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