Solve ((6×2)⁴)⁻⁵: Complex Negative Exponent Expression

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

The power rule states $ (a^m)^n = a^{m \times n} $ because you're applying the exponent n to the entire expression $ a^m $. Think of it as repeated multiplication!

Q: How is this different from multiplying powers with the same base?

Great question! $ a^m \times a^n = a^{m+n} $ (you ADD), but $ (a^m)^n = a^{m \times n} $ (you MULTIPLY). Look for parentheses to identify power-to-a-power situations!

Q: What if I mess up the signs when multiplying?

Be extra careful with negatives! Remember: positive × negative = negative. So $ 4 \times (-5) = -20 $, not $ +20 $. Double-check your multiplication!

Power Rules with Negative Exponents

Insert the corresponding expression:

$\left(\left(6\times2\right)^4\right)^{-5}=$

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Understand the problem

Insert the corresponding expression:

$\left(\left(6\times2\right)^4\right)^{-5}=$

Step-by-step solution

To solve this problem, we'll simplify the expression $\left(\left(6\times2\right)^4\right)^{-5}$ using exponent rules.

Here's a step-by-step breakdown of the solution:

Step 1: Identify the form
The expression is $\left((6 \times 2)^4\right)^{-5}$ . This is a case of the power of a power: $(a^m)^n$ , which can be rewritten as $a^{m \times n}$ .
Step 2: Apply the power of a power rule
Apply the rule: $\left((6 \times 2)^4\right)^{-5} = (6 \times 2)^{4 \times (-5)}$ .
Step 3: Simplify the exponent
Calculate the exponent multiplication: $4 \times (-5) = -20$ . Thus, the expression simplifies to $(6 \times 2)^{-20}$ .

The resulting expression matches the format of choice 3: $\left(6 \times 2\right)^{4\times-5}$ .

Therefore, the correct choice is Choice 3, $\left(6\times2\right)^{4\times-5}$ .

Final Answer

$\left(6\times2\right)^{4\times-5}$

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 $((6\times2)^4)^{-5} = (6\times2)^{4\times(-5)}$
Check: Final exponent should be $4 \times (-5) = -20$ ✓

Common Mistakes

Avoid these frequent errors

Adding or subtracting exponents instead of multiplying
Don't write $(a^m)^n = a^{m+n}$ or $a^{m-n}$ = wrong exponent entirely! This confuses the power rule with multiplication/division rules. Always multiply the exponents when you have a power raised to another power.

Practice Quiz

Test your knowledge with interactive questions

$112^0=\text{?}$

FAQ

Everything you need to know about this question

Why do we multiply the exponents instead of adding them?

The power rule states $(a^m)^n = a^{m \times n}$ because you're applying the exponent n to the entire expression $a^m$ . Think of it as repeated multiplication!

What happens when one exponent is negative?

Negative exponents still follow the same rule! When you multiply $4 \times (-5)$ , you get $-20$ . The negative sign just becomes part of your final exponent.

Do I need to calculate what's inside the parentheses first?

Not necessarily! In this problem, we can apply the power rule to $(6\times2)^{4\times(-5)}$ before calculating $6\times2=12$ . The expression format is what matters.

How is this different from multiplying powers with the same base?

Great question! $a^m \times a^n = a^{m+n}$ (you ADD), but $(a^m)^n = a^{m \times n}$ (you MULTIPLY). Look for parentheses to identify power-to-a-power situations!

What if I mess up the signs when multiplying?

Be extra careful with negatives! Remember: positive × negative = negative. So $4 \times (-5) = -20$ , not $+20$ . Double-check your multiplication!

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