Evaluate (2×9×6)^(-7): Negative Exponent of a Product

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

The negative exponent is about position, not sign! It tells you to flip the base to the denominator. Think of $ x^{-1} = \frac{1}{x} $ - it's positive, just flipped.

Q: Do I need to calculate 2×9×6 first?

No! Keep it as $ (2×9×6) $ in both numerator and denominator. The question asks for the expression, not the numerical answer.

Q: What's the difference between the exponent being negative and the answer being negative?

A negative exponent creates a fraction: $ a^{-n} = \frac{1}{a^n} $. A negative answer would have a minus sign in front of the whole expression.

Q: Can I move the negative exponent to the denominator?

That's exactly what you should do! $ (2×9×6)^{-7} = \frac{1}{(2×9×6)^7} $. Moving to the denominator makes the exponent positive.

Q: What if the base was already in a fraction?

If you had $ \left(\frac{a}{b}\right)^{-n} $, it would become $ \left(\frac{b}{a}\right)^{n} $ - the fraction flips and the exponent becomes positive!

Negative Exponents with Product Expressions

Choose the expression that corresponds to the following:

$\left(2\times9\times6\right)^{-7}=$

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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 According to the exponent laws, when we have a negative exponent

00:08 We can convert to the reciprocal number and obtain a positive exponent

00:12 We will apply this formula to our exercise

00:17 We'll write the reciprocal number (1 divided by the number)

00:21 Proceed to raise to the positive exponent

00:24 This is the solution

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Choose the expression that corresponds to the following:

$\left(2\times9\times6\right)^{-7}=$

Step-by-step solution

We first need to apply the exponent rule for powers with negative exponents, specifically the rule for the power of a product which states that:
$\left(a \times b \times c \right)^{-n} = \frac{1}{(a \times b \times c)^n}$ .

In this problem, we have three numbers multiplied inside the parentheses: 2, 9, and 6. The exponent is -7.

By applying the power of a product rule with a negative exponent here, we get:
$\left(2\times9\times6\right)^{-7} = \frac{1}{\left(2\times9\times6\right)^7}$ .

Therefore the correct answer is:
$\frac{1}{\left(2\times9\times6\right)^7}$

Final Answer

$\frac{1}{\left(2\times9\times6\right)^7}$

Key Points to Remember

Essential concepts to master this topic

Rule: A negative exponent means take the reciprocal
Technique: $(abc)^{-n} = \frac{1}{(abc)^n}$ flips to denominator
Check: Negative exponent becomes positive in denominator: $(108)^{-7} = \frac{1}{(108)^7}$ ✓

Common Mistakes

Avoid these frequent errors

Making the entire expression negative
Don't think $(2×9×6)^{-7}$ equals $-(2×9×6)^7$ = negative result! The negative exponent doesn't make the answer negative, it creates a reciprocal. Always remember: negative exponent means flip to denominator, not make negative.

Practice Quiz

Test your knowledge with interactive questions

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

FAQ

Everything you need to know about this question

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

The negative exponent is about position, not sign! It tells you to flip the base to the denominator. Think of $x^{-1} = \frac{1}{x}$ - it's positive, just flipped.

Do I need to calculate 2×9×6 first?

No! Keep it as $(2×9×6)$ in both numerator and denominator. The question asks for the expression, not the numerical answer.

What's the difference between the exponent being negative and the answer being negative?

A negative exponent creates a fraction: $a^{-n} = \frac{1}{a^n}$ . A negative answer would have a minus sign in front of the whole expression.

Can I move the negative exponent to the denominator?

That's exactly what you should do! $(2×9×6)^{-7} = \frac{1}{(2×9×6)^7}$ . Moving to the denominator makes the exponent positive.

What if the base was already in a fraction?

If you had $\left(\frac{a}{b}\right)^{-n}$ , it would become $\left(\frac{b}{a}\right)^{n}$ - the fraction flips and the exponent becomes positive!

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