Solve (9×2)^(-6): Negative Exponent Expression

Q: Why does the negative exponent apply to both 9 and 2?

The power of a product rule says that when you raise a product to any power, you raise each factor to that power. Since -6 is the exponent for the entire product $ (9 \times 2) $, it applies to both numbers.

Q: Does a negative exponent make the answer negative?

No! A negative exponent means reciprocal, not negative. $ (9 \times 2)^{-6} = \frac{1}{(9 \times 2)^6} $, which is a positive fraction, not a negative number.

Q: What if I get confused about which rule to use?

Look for the structure! When you see $ (a \times b)^n $ or $ (a \cdot b)^n $, use the power of a product rule. The exponent goes to each factor in the multiplication.

Q: How do I remember that exponents distribute over multiplication?

Think of it as repeated multiplication: $ (9 \times 2)^3 = (9 \times 2)(9 \times 2)(9 \times 2) = 9^3 \times 2^3 $. This pattern works for any exponent, including negative ones!

Power of Product Rule with Negative Exponents

Insert the corresponding expression:

$\left(9\times2\right)^{-6}=$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:07 Let's simplify the problem together.

00:11 To handle parentheses with multiplication and exponents,

00:15 raise each number inside to the power outside.

00:20 Now, let's apply this rule to our problem.

00:28 And there you have it! That's the solution.

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Insert the corresponding expression:

$\left(9\times2\right)^{-6}=$

Step-by-step solution

To solve this problem, we'll apply the power of a product rule for exponents:

Step 1: Identify the expression $(9 \times 2)^{-6}$ .
Step 2: Apply the power of a product rule, which states $(a \times b)^n = a^n \times b^n$ .
Step 3: Simplify the expression using the rule $(9 \times 2)^{-6} = 9^{-6} \times 2^{-6}$ .

By performing these steps, the expression $(9 \times 2)^{-6}$ becomes equivalent to $9^{-6} \times 2^{-6}$ .

Therefore, the solution to the problem is $9^{-6} \times 2^{-6}$ .

Final Answer

$9^{-6}\times2^{-6}$

Key Points to Remember

Essential concepts to master this topic

Rule: Apply $(a \times b)^n = a^n \times b^n$ for any exponent
Technique: Distribute the exponent -6 to both factors: $9^{-6} \times 2^{-6}$
Check: Both expressions equal $\frac{1}{(9 \times 2)^6} = \frac{1}{18^6}$ ✓

Common Mistakes

Avoid these frequent errors

Treating negative exponents as multiplication by -1
Don't change $(9 \times 2)^{-6}$ to $-(9 \times 2)^6$ = wrong sign! Negative exponents mean reciprocal, not negative numbers. Always remember that $a^{-n} = \frac{1}{a^n}$ , so apply the power rule first.

Practice Quiz

Test your knowledge with interactive questions

Choose the expression that corresponds to the following:

$$ $\left(2\times11\right)^5=$

FAQ

Everything you need to know about this question

Why does the negative exponent apply to both 9 and 2?

The power of a product rule says that when you raise a product to any power, you raise each factor to that power. Since -6 is the exponent for the entire product $(9 \times 2)$ , it applies to both numbers.

Does a negative exponent make the answer negative?

No! A negative exponent means reciprocal, not negative. $(9 \times 2)^{-6} = \frac{1}{(9 \times 2)^6}$ , which is a positive fraction, not a negative number.

Can I simplify 9×2 to 18 first?

You could, but the question asks for the equivalent expression using the power of a product rule. The goal is to show how the exponent distributes: $(9 \times 2)^{-6} = 9^{-6} \times 2^{-6}$ .

What if I get confused about which rule to use?

Look for the structure! When you see $(a \times b)^n$ or $(a \cdot b)^n$ , use the power of a product rule. The exponent goes to each factor in the multiplication.

How do I remember that exponents distribute over multiplication?

Think of it as repeated multiplication: $(9 \times 2)^3 = (9 \times 2)(9 \times 2)(9 \times 2) = 9^3 \times 2^3$ . This pattern works for any exponent, including negative ones!

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