Calculate (12×3)^5: Evaluating the Fifth Power of a Product

Q: Why can't I just multiply 12 × 3 first to get 36⁵?

You absolutely can! $(12 \times 3)^5 = 36^5$ is correct. However, the question asks for the distributed form, which is $12^5 \times 3^5$. Both expressions are equal but show different mathematical approaches.

Q: When do I use the power of a product rule?

Use this rule whenever you see multiplication inside parentheses raised to a power. The rule $(a \times b)^n = a^n \times b^n$ helps you distribute the exponent to each factor separately.

Q: Does this work with more than two factors?

Yes! For example, $(2 \times 3 \times 5)^4 = 2^4 \times 3^4 \times 5^4$. The exponent distributes to every single factor inside the parentheses.

Q: What if I have division inside the parentheses?

The same rule applies! $\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$. The exponent distributes to both the numerator and denominator.

Q: How is this different from (a + b)ⁿ?

Big difference! Addition doesn't distribute like multiplication. $(a + b)^n \neq a^n + b^n$. Only use the distribution rule with multiplication and division, not addition or subtraction.

Power of a Product with Exponent Rules

Choose the expression that corresponds to the following:

$\left(12\times3\right)^5=$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Simplify the following problem

00:03 In order to expand parentheses containing a multiplication operation with an outside exponent

00:06 Raise each factor to the power

00:13 We will apply this formula to our exercise

00:20 This is the solution

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Choose the expression that corresponds to the following:

$\left(12\times3\right)^5=$

Step-by-step solution

The given expression is $(12 \times 3)^5$ . The power of a product rule states that $(a \times b)^n = a^n \times b^n$ . We will apply this formula to the expression.

Firstly, identify the base of the power as the product $12 \times 3$ .
Secondly, recognize that the exponent applied to this product is 5.
According to the rule, the power of a product can be distributed to each factor in the product, which means: $(12 \times 3)^5 = 12^5 \times 3^5$ .

Therefore, the expression $(12 \times 3)^5$ corresponds to $12^5 \times 3^5$ .

Final Answer

$12^5\times3^5$

Key Points to Remember

Essential concepts to master this topic

Power Rule: Distribute exponents to each factor in the product
Technique: $(12 \times 3)^5 = 12^5 \times 3^5$ using distributive property
Check: Verify both sides equal the same value: $36^5 = 12^5 \times 3^5$ ✓

Common Mistakes

Avoid these frequent errors

Applying exponent to only one factor
Don't write $(12 \times 3)^5 = 12^5 \times 3$ = wrong answer! This ignores the power rule and leaves one factor without the exponent. Always distribute the exponent to every factor inside the parentheses.

Practice Quiz

Test your knowledge with interactive questions

$112^0=\text{?}$

FAQ

Everything you need to know about this question

Why can't I just multiply 12 × 3 first to get 36⁵?

You absolutely can! $(12 \times 3)^5 = 36^5$ is correct. However, the question asks for the distributed form, which is $12^5 \times 3^5$ . Both expressions are equal but show different mathematical approaches.

When do I use the power of a product rule?

Use this rule whenever you see multiplication inside parentheses raised to a power. The rule $(a \times b)^n = a^n \times b^n$ helps you distribute the exponent to each factor separately.

Does this work with more than two factors?

Yes! For example, $(2 \times 3 \times 5)^4 = 2^4 \times 3^4 \times 5^4$ . The exponent distributes to every single factor inside the parentheses.

What if I have division inside the parentheses?

The same rule applies! $\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$ . The exponent distributes to both the numerator and denominator.

How is this different from (a + b)ⁿ?

Big difference! Addition doesn't distribute like multiplication. $(a + b)^n \neq a^n + b^n$ . Only use the distribution rule with multiplication and division, not addition or subtraction.

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