Solve Expression: ((5×3)^4)^(-3) Using Power Rules

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

The power of a power rule says $ (a^m)^n = a^{m \times n} $. Adding exponents only works when multiplying powers with the same base, like $ a^m \cdot a^n = a^{m+n} $.

Q: What does the negative exponent mean?

A negative exponent means "flip and make positive". So $ a^{-n} = \frac{1}{a^n} $. In our problem, $ (5\times3)^{-12} = \frac{1}{(5\times3)^{12}} $.

Q: Should I calculate 5 × 3 = 15 first?

You could, but it's not necessary! The power rules work the same way whether you have $ (15)^{-12} $ or $ (5\times3)^{-12} $. Keep it as $ (5\times3) $ to match the answer choices.

Q: How do I remember which rule to use?

Look at the structure! If you see parentheses raised to a power like $ (something)^{power} $, use the power of a power rule and multiply exponents.

Q: What if I forget to handle the negative exponent?

You'll get the reciprocal of the right answer! Always remember: negative exponent = flip to make a fraction. Check by seeing if your answer makes sense with the original expression.

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

1

Understand the problem

Insert the corresponding expression:

$\left(\left(5\times3\right)^4\right)^{-3}=$

2

Step-by-step solution

To solve the problem, let us simplify the expression $\left(\left(5\times3\right)^4\right)^{-3}$ .

First, recognize that the expression inside the parentheses, $5 \times 3$ , can be multiplied to give us 15. However, we'll focus on exponent rules directly.

Step 1: Apply the Power of a Power Rule.
Using the formula $(a^m)^n = a^{m \times n}$ , simplification gives: $\left(\left(5\times3\right)^4\right)^{-3} = \left( (5 \times 3)^{4 \times -3} \right) = \left(5\times3\right)^{-12}.$
Step 2: Apply the Negative Exponent Rule.
Using $a^{-n} = \frac{1}{a^n}$ , the expression becomes: $\left(5\times3\right)^{-12} = \frac{1}{\left(5\times3\right)^{12}}.$

Therefore, the solution to the given expression is $\frac{1}{\left(5\times3\right)^{12}}$ .

Now, let's verify the answer with the choices provided:

Choice 1: $\frac{1}{\left(5\times3\right)^{12}}$ - This matches our solution.
Choice 2: $(5\times3)^{12}$ - Incorrect, doesn't account for the negative exponent.
Choice 3: $\frac{1}{\left(5\times3\right)^{-1}}$ - Incorrect power and doesn't represent the full expression.
Choice 4: $\frac{1}{\left(5\times3\right)^1}$ - Incorrect power and doesn't reflect the original problem.

Thus, the correct choice is Choice 1: $\frac{1}{\left(5\times3\right)^{12}}$ .

3

Final Answer

$\frac{1}{\left(5\times3\right)^{12}}$

Key Points to Remember

Essential concepts to master this topic

Power Rule: When raising a power to another power, multiply the exponents
Technique: $(a^m)^n = a^{m \times n}$ , so $((5\times3)^4)^{-3} = (5\times3)^{4 \times (-3)} = (5\times3)^{-12}$
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 4 + (-3) = 1 to get $(5\times3)^1$ ! This confuses the power rule with the product rule and gives completely wrong results. Always multiply exponents when raising a power to another power: 4 × (-3) = -12.

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 of a power rule says $(a^m)^n = a^{m \times n}$ . Adding exponents only works when multiplying powers with the same base, like $a^m \cdot a^n = a^{m+n}$ .

What does the negative exponent mean?

+

A negative exponent means "flip and make positive". So $a^{-n} = \frac{1}{a^n}$ . In our problem, $(5\times3)^{-12} = \frac{1}{(5\times3)^{12}}$ .

Should I calculate 5 × 3 = 15 first?

+

You could, but it's not necessary! The power rules work the same way whether you have $(15)^{-12}$ or $(5\times3)^{-12}$ . Keep it as $(5\times3)$ to match the answer choices.

How do I remember which rule to use?

+

Look at the structure! If you see parentheses raised to a power like $(something)^{power}$ , use the power of a power rule and multiply exponents.

What if I forget to handle the negative exponent?

+

You'll get the reciprocal of the right answer! Always remember: negative exponent = flip to make a fraction. Check by seeing if your answer makes sense with the original expression.

🌟 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

Solve Expression: ((5×3)^4)^(-3) Using Power Rules

Power Rules with Negative Exponents

❤️ Continue Your Math Journey!