Evaluate ((7×4)^-6)^5: Complex Exponent Expression

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

The power of a power rule says $ (a^m)^n = a^{m \times n} $. Think of it as: you're multiplying the base by itself m times, and then doing that whole process n times, so you multiply m × n!

Q: What's the difference between this and adding exponents?

You add exponents when multiplying same bases: $ a^m \times a^n = a^{m+n} $. You multiply exponents when raising a power to a power: $ (a^m)^n = a^{m \times n} $. Different operations, different rules!

Q: Do I need to calculate 7×4 first?

No! Keep $ (7\times4) $ as one unit. The question asks for the expression form, not the final numerical answer. Focus on applying the exponent rules correctly.

Q: Why is the answer negative?

Because $ -6 \times 5 = -30 $. When you multiply a negative exponent by a positive number, you get a negative result. The negative sign stays in the final exponent.

Q: How can I remember which rule to use?

Parentheses around a power? Multiply exponents: $ (a^m)^n = a^{m \times n} $Same bases being multiplied? Add exponents: $ a^m \times a^n = a^{m+n} $

Power Rules with Nested Exponents

Insert the corresponding expression:

$\left(\left(7\times4\right)^{-6}\right)^5=$

❤️ Continue Your Math Journey!

We have hundreds of course questions with personalized recommendations + Account 100% premium

Step-by-step video solution

Watch the teacher solve the problem with clear explanations

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Insert the corresponding expression:

$\left(\left(7\times4\right)^{-6}\right)^5=$

Step-by-step solution

To simplify the expression $\left(\left(7\times4\right)^{-6}\right)^5$ , follow these steps, checking against the choices provided:

Step 1: Apply the power of a power rule.

The expression inside the parentheses $7\times4$ acts as a single term $a$ .
Therefore, by applying $(a^m)^n = a^{m \times n}$ , we simplify:

$\left(\left(7\times4\right)^{-6}\right)^5 = \left(7\times4\right)^{-6 \times 5}$

Step 2: Multiply the exponents.

Calculate $-6 \times 5 = -30$ .
Hence, the expression simplifies to $\left(7\times4\right)^{-30}$ .

Conclusion:

The correct simplified form of the expression is $\left(7\times4\right)^{-6\times5}$ , aligning with choice 2 and your provided correct answer.

Final Answer

$\left(7\times4\right)^{-6\times5}$

Key Points to Remember

Essential concepts to master this topic

Power Rule: When raising a power to a power, multiply exponents
Technique: Apply $(a^m)^n = a^{m \times n}$ so $-6 \times 5 = -30$
Check: Final answer should be $(7\times4)^{-30}$ which matches choice 2 ✓

Common Mistakes

Avoid these frequent errors

Adding or subtracting exponents instead of multiplying
Don't add exponents like -6 + 5 = -1 or subtract like -6 - 5 = -11! This confuses the power rule with addition/subtraction rules and gives completely wrong results. Always multiply exponents when raising a power to a power using $(a^m)^n = a^{m \times n}$ .

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 says $(a^m)^n = a^{m \times n}$ . Think of it as: you're multiplying the base by itself m times, and then doing that whole process n times, so you multiply m × n!

What's the difference between this and adding exponents?

You add exponents when multiplying same bases: $a^m \times a^n = a^{m+n}$ . You multiply exponents when raising a power to a power: $(a^m)^n = a^{m \times n}$ . Different operations, different rules!

Do I need to calculate 7×4 first?

No! Keep $(7\times4)$ as one unit. The question asks for the expression form, not the final numerical answer. Focus on applying the exponent rules correctly.

Why is the answer negative?

Because $-6 \times 5 = -30$ . When you multiply a negative exponent by a positive number, you get a negative result. The negative sign stays in the final exponent.

How can I remember which rule to use?

Parentheses around a power? Multiply exponents: $(a^m)^n = a^{m \times n}$
Same bases being multiplied? Add exponents: $a^m \times a^n = a^{m+n}$

🌟 Unlock Your Math Potential

Get unlimited access to all 18 Exponents Rules questions, detailed video solutions, and personalized progress tracking.

📹

Unlimited Video Solutions

Step-by-step explanations for every problem

📊

Progress Analytics

Track your mastery across all topics

🚫

Ad-Free Learning

Focus on math without distractions

No credit card required • Cancel anytime