Solve Nested Exponents: Calculate ((15×3)^10)^10

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

The power of a power rule states $ (a^m)^n = a^{m×n} $. We add exponents only when multiplying powers with the same base, like $ a^m \times a^n = a^{m+n} $.

Q: Do I need to calculate 15 × 3 first?

No! Leave it as $ (15×3) $ since the problem asks for the expression form, not the numerical value. The focus is on applying the exponent rule correctly.

Q: What if there were three nested exponents like (((15×3)^2)^3)^4?

Apply the power rule step by step: $ (((15×3)^2)^3)^4 = ((15×3)^{2×3})^4 = (15×3)^{6×4} = (15×3)^{24} $. Or multiply all exponents at once: 2 × 3 × 4 = 24.

Q: How can I remember when to multiply vs. add exponents?

Use this trick: Power to a power = multiply, Same base multiplication = add. Look for parentheses - they usually signal the power of a power rule!

Q: Why is (15×3)^20 wrong?

Because 10 + 10 = 20 uses the wrong rule. Adding exponents applies to $ a^m \times a^n $, but we have $ (a^m)^n $ which requires multiplication: 10 × 10 = 100.

Power Rules with Nested Exponents

Insert the corresponding expression:

$\left(\right.\left(15\times3\right)^{10})^{10}=$

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Understand the problem

Insert the corresponding expression:

$\left(\right.\left(15\times3\right)^{10})^{10}=$

Step-by-step solution

To solve this problem, we'll follow these steps:

Step 1: Identify the given expression.
Step 2: Apply the power of a power rule using exponent multiplication.
Step 3: Confirm the result against the given choices.

Now, let's work through each step:

Step 1: Identify the Given Expression.
The given expression is $\left((15 \times 3)^{10}\right)^{10}$ .

Step 2: Apply the Power of a Power Rule.
According to the exponent rule, $(a^m)^n = a^{m \times n}$ , we multiply the exponents:

The base is $15 \times 3$ .
The inner exponent is 10, and the outer exponent is 10.
So, we apply the rule: $((15 \times 3)^{10})^{10} = (15 \times 3)^{10 \times 10}$ .
This simplifies to $(15 \times 3)^{100}$ .

Step 3: Confirm the Result Against the Choices.
The expression simplifies to $(15 \times 3)^{100}$ .

The correct choice from the options provided is:

$\left(15\times3\right)^{100}$

Final Answer

$\left(15\times3\right)^{100}$

Key Points to Remember

Essential concepts to master this topic

Power Rule: When raising a power to a power, multiply the exponents
Technique: $((15×3)^{10})^{10} = (15×3)^{10×10} = (15×3)^{100}$
Check: Verify exponent multiplication: 10 × 10 = 100 ✓

Common Mistakes

Avoid these frequent errors

Adding exponents instead of multiplying
Don't add the exponents like 10 + 10 = 20 to get (15×3)^20! This uses the wrong rule and gives an incorrect answer. Always multiply exponents when you have a power raised to another power: (a^m)^n = 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 we multiply the exponents instead of adding them?

The power of a power rule states $(a^m)^n = a^{m×n}$ . We add exponents only when multiplying powers with the same base, like $a^m \times a^n = a^{m+n}$ .

Do I need to calculate 15 × 3 first?

No! Leave it as $(15×3)$ since the problem asks for the expression form, not the numerical value. The focus is on applying the exponent rule correctly.

What if there were three nested exponents like (((15×3)^2)^3)^4?

Apply the power rule step by step: $(((15×3)^2)^3)^4 = ((15×3)^{2×3})^4 = (15×3)^{6×4} = (15×3)^{24}$ . Or multiply all exponents at once: 2 × 3 × 4 = 24.

How can I remember when to multiply vs. add exponents?

Use this trick: Power to a power = multiply, Same base multiplication = add. Look for parentheses - they usually signal the power of a power rule!

Why is (15×3)^20 wrong?

Because 10 + 10 = 20 uses the wrong rule. Adding exponents applies to $a^m \times a^n$ , but we have $(a^m)^n$ which requires multiplication: 10 × 10 = 100.

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