Solve (2×3)²)⁵: Evaluating Nested Exponential Expressions

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

The power of power rule says $ (a^m)^n = a^{m×n} $. Think of it this way: $ ((2×3)^2)^5 $ means you multiply $ (2×3)^2 $ by itself 5 times, which gives you 2×5=10 total factors of (2×3).

Q: When do I add exponents versus multiply them?

Add exponents when multiplying same bases: $ a^m × a^n = a^{m+n} $. Multiply exponents when raising a power to a power: $ (a^m)^n = a^{m×n} $. Different situations, different rules!

Q: What if I calculated the base first, like 2×3=6?

You could do that! $ ((2×3)^2)^5 = (6^2)^5 = 6^{10} $. But the question asks for the form with $ (2×3) $ kept as the base, so $ (2×3)^{2×5} $ is the expected answer.

Q: How can I remember the power of power rule?

Think "multiply to multiply" - when you have nested exponents (powers of powers), you multiply them together. Or remember: parentheses around an exponent means multiply that exponent by the outside power.

Q: What's the difference between this and (2×3)² × (2×3)⁵?

Great question! $ (2×3)^2 × (2×3)^5 = (2×3)^{2+5} = (2×3)^7 $ (add exponents when multiplying same bases). But $ ((2×3)^2)^5 = (2×3)^{2×5} = (2×3)^{10} $ (multiply exponents for power of power).

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:

$\left(\right.\left(2\times3\right)^2)^5=$

2

Step-by-step solution

To solve the problem, we will simplify the expression $\left(\left(2 \times 3\right)^2\right)^5$ using the power of a power exponent rule. Follow these steps:

Step 1: Identify the form of the expression. The given expression is $\left(\left(2 \times 3\right)^2\right)^5$ .
Step 2: Apply the power of a power rule, which states that $(a^m)^n = a^{m \times n}$ .
Step 3: Here, the base is $2 \times 3$ , the first exponent ( $m$ ) is 2, and the second exponent ( $n$ ) is 5.
Step 4: Multiply the exponents: $2 \times 5 = 10$ .

Therefore, the expression simplifies to $\left(2 \times 3\right)^{10}$ . However, for the purpose of matching the form requested, it can be expressed as $\left(2 \times 3\right)^{2 \times 5}$ .

Next, we evaluate the given choices:

Choice 1: $\left(2 \times 3\right)^{2+5}$ — This incorrectly adds the exponents instead of multiplying them.
Choice 2: $\left(2 \times 3\right)^{5-2}$ — This incorrectly subtracts the exponents.
Choice 3: $\left(2 \times 3\right)^{2\times5}$ — This correctly multiplies the exponents, which we found is the right simplification.
Choice 4: $\left(2 \times 3\right)^{\frac{5}{2}}$ — This introduces division of exponents, which is not applicable here.

The correct choice is Choice 3: $\left(2 \times 3\right)^{2\times5}$ .

3

Final Answer

$\left(2\times3\right)^{2\times5}$

Key Points to Remember

Essential concepts to master this topic

Power of Power Rule: When raising a power to another power, multiply exponents
Technique: $(a^m)^n = a^{m \times n}$ so $((2×3)^2)^5 = (2×3)^{2×5}$
Check: Verify by expanding: $2×5 = 10$ gives $(2×3)^{10}$ ✓

Common Mistakes

Avoid these frequent errors

Adding exponents instead of multiplying them
Don't add exponents like $((2×3)^2)^5 = (2×3)^{2+5}$ = wrong answer! This confuses the power-of-power rule with the product rule. Always multiply the exponents when you have a power raised to another power.

Practice Quiz

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$112^0=\text{?}$

FAQ

Everything you need to know about this question

Why do I multiply the exponents instead of adding them?

+

The power of power rule says $(a^m)^n = a^{m×n}$ . Think of it this way: $((2×3)^2)^5$ means you multiply $(2×3)^2$ by itself 5 times, which gives you 2×5=10 total factors of (2×3).

When do I add exponents versus multiply them?

+

Add exponents when multiplying same bases: $a^m × a^n = a^{m+n}$ . Multiply exponents when raising a power to a power: $(a^m)^n = a^{m×n}$ . Different situations, different rules!

What if I calculated the base first, like 2×3=6?

+

You could do that! $((2×3)^2)^5 = (6^2)^5 = 6^{10}$ . But the question asks for the form with $(2×3)$ kept as the base, so $(2×3)^{2×5}$ is the expected answer.

How can I remember the power of power rule?

+

Think "multiply to multiply" - when you have nested exponents (powers of powers), you multiply them together. Or remember: parentheses around an exponent means multiply that exponent by the outside power.

What's the difference between this and (2×3)² × (2×3)⁵?

+

Great question! $(2×3)^2 × (2×3)^5 = (2×3)^{2+5} = (2×3)^7$ (add exponents when multiplying same bases). But $((2×3)^2)^5 = (2×3)^{2×5} = (2×3)^{10}$ (multiply exponents for power of power).

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Solve (2×3)²)⁵: Evaluating Nested Exponential Expressions

Exponent Rules with Nested Power Expressions

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