Solve (6×5)^(-8))^(-4): Compound Negative Exponents Problem

Q: Why do I multiply the exponents instead of adding them?

The power of a power rule states $ (a^m)^n = a^{m \times n} $. You only add exponents when multiplying terms with the same base, like $ a^2 \cdot a^3 = a^5 $.

Q: What happens when I multiply two negative numbers?

When you multiply two negative numbers, the result is positive! So $ -8 \times (-4) = +32 $. Remember: negative × negative = positive.

Q: Can I simplify 6×5 first before applying the exponents?

You could calculate $ 6 \times 5 = 30 $ first, giving you $ 30^{32} $. However, the question asks for the form $ (6×5)^{32} $, so keep it as is!

Q: How do I remember the power rule?

Think of it as "power to a power means multiply". If you see parentheses with an exponent outside, like $ (something^a)^b $, always multiply the exponents: $ a \times b $.

Q: What if one of the exponents was positive?

The rule stays the same! For example, $ (a^{-8})^4 = a^{-8 \times 4} = a^{-32} $. Always multiply the exponents, whether they're positive or negative.

Power Rules with Double Negative Exponents

Insert the corresponding expression:

$\left(\left(6\times5\right)^{-8}\right)^{-4}=$

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Understand the problem

Insert the corresponding expression:

$\left(\left(6\times5\right)^{-8}\right)^{-4}=$

Step-by-step solution

To solve this problem, we will follow these steps:

Step 1: Apply the power of a power rule.
Step 2: Simplify the resulting power.

Let's work through the process:

Step 1: Apply the power of a power rule, which states that $\left(a^m\right)^n = a^{m \times n}$ .
We have $\left(\left(6\times5\right)^{-8}\right)^{-4}$ . We can rewrite this using the power of a power rule:

$\left(\left(6\times5\right)^{-8}\right)^{-4} = \left(6\times5\right)^{-8 \times (-4)} = \left(6\times5\right)^{32}$

Step 2: By calculating the exponent: $-8 \times (-4) = 32$ , we find the final simplified expression to be $\left(6\times5\right)^{32}$ .

Therefore, the expression reduces to $\left(6\times5\right)^{32}$ , which matches choice 2.

Final Answer

$\left(6\times5\right)^{32}$

Key Points to Remember

Essential concepts to master this topic

Power Rule: When raising a power to a power, multiply the exponents
Technique: $(a^m)^n = a^{m \times n}$ , so $-8 \times (-4) = 32$
Check: Two negative signs multiply to positive: $(-8) \times (-4) = +32$ ✓

Common Mistakes

Avoid these frequent errors

Adding exponents instead of multiplying
Don't add exponents like -8 + (-4) = -12! This gives $(6×5)^{-12}$ which is completely wrong. The power of a power rule requires multiplication, not addition. Always multiply the exponents: $-8 \times (-4) = 32$ .

Practice Quiz

Test your knowledge with interactive questions

$112^0=\text{?}$

FAQ

Everything you need to know about this question

Why do I multiply the exponents instead of adding them?

The power of a power rule states $(a^m)^n = a^{m \times n}$ . You only add exponents when multiplying terms with the same base, like $a^2 \cdot a^3 = a^5$ .

What happens when I multiply two negative numbers?

When you multiply two negative numbers, the result is positive! So $-8 \times (-4) = +32$ . Remember: negative × negative = positive.

Can I simplify 6×5 first before applying the exponents?

You could calculate $6 \times 5 = 30$ first, giving you $30^{32}$ . However, the question asks for the form $(6×5)^{32}$ , so keep it as is!

How do I remember the power rule?

Think of it as "power to a power means multiply". If you see parentheses with an exponent outside, like $(something^a)^b$ , always multiply the exponents: $a \times b$ .

What if one of the exponents was positive?

The rule stays the same! For example, $(a^{-8})^4 = a^{-8 \times 4} = a^{-32}$ . Always multiply the exponents, whether they're positive or negative.

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