Solve ((7×6)^-5)^3: Complex Exponent Expression Evaluation

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

The power of a power rule says $ (a^m)^n = a^{m×n} $. Think of it as: you're multiplying the base by itself m times, and then doing that entire process n times, which gives you m×n total multiplications.

Q: What's the difference between negative exponents in the base vs. the final answer?

A negative exponent in the base like $ (7×6)^{-5} $ stays negative when multiplied. The final $ (7×6)^{-15} $ converts to $ \frac{1}{(7×6)^{15}} $ because any negative exponent means reciprocal.

Q: Should I calculate 7×6 = 42 first?

Not necessary! The problem asks for the expression form, not the numerical value. Keeping it as $ (7×6) $ matches the answer choices perfectly.

Q: How can I remember the power rule?

Think "power to power = multiply". It's like saying "do this 5 times, and repeat that whole thing 3 times" = 5×3 = 15 times total.

Q: What if I see a positive exponent in the final answer?

Double-check your multiplication! $ (-5) × 3 = -15 $, which stays negative. A positive result would mean you made a sign error in your calculation.

Step-by-step video solution

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Step-by-step written solution

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1

Understand the problem

Insert the corresponding expression:

$\left(\left(7\times6\right)^{-5}\right)^3=$

2

Step-by-step solution

To solve this problem, we must simplify the expression $\left(\left(7 \times 6\right)^{-5}\right)^3$ .

We'll follow these steps:

Step 1: Apply the power of a power rule.
Step 2: Simplify the expression into a single exponent.
Step 3: Convert the negative exponent to a fraction form.

Now, let's work through each step:

Step 1: The expression $\left(7 \times 6\right)^{-5}$ is raised to the power 3. By the power of a power rule, we multiply the exponents:

\left((7 \times 6)^{-5}\right)^3 = (7 \times 6)^{-5 \times 3} = (7 \times 6)^{-15}

Step 2: This simplifies the expression to $(7 \times 6)^{-15}$ .

Step 3: Since we have a negative exponent, we convert it to a fraction:

(7 \times 6)^{-15} = \frac{1}{(7 \times 6)^{15}}

Therefore, the simplified expression is:

\frac{1}{\left(7 \times 6\right)^{15}}

Comparing this result with the given choices, the correct answer is:

- Choice 3:

\frac{1}{\left(7 \times 6\right)^{15}}

The other choices are incorrect because they either have the wrong exponent or incorrectly handle the negative exponent.

Thus, the correct answer to the problem is $\frac{1}{\left(7 \times 6\right)^{15}}$ .

3

Final Answer

$\frac{1}{\left(7\times6\right)^{15}}$

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 (-5) × 3 = -15
Check: Negative exponent means reciprocal: $a^{-n} = \frac{1}{a^n}$ ✓

Common Mistakes

Avoid these frequent errors

Adding exponents instead of multiplying
Don't add exponents like (-5) + 3 = -2! This gives $\frac{1}{(7×6)^2}$ which is completely wrong. The power rule requires multiplication: when you have $(a^m)^n$ , always multiply the exponents to get $a^{m×n}$ .

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 $(a^m)^n = a^{m×n}$ . Think of it as: you're multiplying the base by itself m times, and then doing that entire process n times, which gives you m×n total multiplications.

What's the difference between negative exponents in the base vs. the final answer?

+

A negative exponent in the base like $(7×6)^{-5}$ stays negative when multiplied. The final $(7×6)^{-15}$ converts to $\frac{1}{(7×6)^{15}}$ because any negative exponent means reciprocal.

Should I calculate 7×6 = 42 first?

+

Not necessary! The problem asks for the expression form, not the numerical value. Keeping it as $(7×6)$ matches the answer choices perfectly.

How can I remember the power rule?

+

Think "power to power = multiply". It's like saying "do this 5 times, and repeat that whole thing 3 times" = 5×3 = 15 times total.

What if I see a positive exponent in the final answer?

+

Double-check your multiplication! $(-5) × 3 = -15$ , which stays negative. A positive result would mean you made a sign error in your calculation.

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Solve ((7×6)^-5)^3: Complex Exponent Expression Evaluation

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