Calculate the Product: Solving 3^4 × 4^4 Expression

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

You can use $ a^n \times b^n = (a \times b)^n $ only when the exponents are identical. If the exponents differ, you must calculate each power separately first.

Q: What if the bases were the same instead?

If you had $ 3^4 \times 3^4 $, then you'd add the exponents: $ 3^4 \times 3^4 = 3^{4+4} = 3^8 $. Same base = add exponents!

Q: Can I use this rule with more than two numbers?

Absolutely! For example: $ 2^3 \times 5^3 \times 7^3 = (2 \times 5 \times 7)^3 = 70^3 $. As long as all exponents match, combine all the bases.

Q: Is it easier to calculate 12^4 or 3^4 × 4^4?

It depends on the numbers! Sometimes $ 12^4 $ is easier, but for larger numbers, calculating $ 3^4 = 81 $ and $ 4^4 = 256 $ separately might be simpler.

Exponent Rules with Product Properties

Choose the expression that corresponds to the following:

$3^4\times4^4=$

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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 When we are presented with a multiplication where each factor has the same exponent (N)

00:07 The entire multiplication can be written with the exponent (N)

00:10 We will apply this formula to our exercise

00:18 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:

$3^4\times4^4=$

Step-by-step solution

To solve the given expression, we need to apply the 'Power of a Product' rule in exponentiation. This rule states that for any numbers $a$ and $b$ :

$a^n \times b^n = (a \times b)^n$

In this problem, the base numbers are 3 and 4, and the exponent is 4. Therefore, we can rewrite the expression $3^4 \times 4^4$ using the power of a product rule like so:

Identify the bases: 3 and 4.
Identify the common exponent: 4.
Apply the rule: $(3 \times 4)^4$ .

Thus, the expression $3^4 \times 4^4$ can be rewritten as $(3 \times 4)^4$ .

Final Answer

$\left(3\times4\right)^4$

Key Points to Remember

Essential concepts to master this topic

Power Rule: When bases differ but exponents match, use $a^n \times b^n = (a \times b)^n$
Technique: Combine bases first: $3^4 \times 4^4 = (3 \times 4)^4 = 12^4$
Check: Calculate both ways: $3^4 = 81, 4^4 = 256$ , so $81 \times 256 = 20,736 = 12^4$ ✓

Common Mistakes

Avoid these frequent errors

Adding exponents when multiplying different bases
Don't think $3^4 \times 4^4 = (3 \times 4)^{4+4} = 12^8$ ! Adding exponents only works when the bases are the same. Always check if bases match before applying any exponent rule.

Practice Quiz

Test your knowledge with interactive questions

$112^0=\text{?}$

FAQ

Everything you need to know about this question

When can I use the power of a product rule?

You can use $a^n \times b^n = (a \times b)^n$ only when the exponents are identical. If the exponents differ, you must calculate each power separately first.

What if the bases were the same instead?

If you had $3^4 \times 3^4$ , then you'd add the exponents: $3^4 \times 3^4 = 3^{4+4} = 3^8$ . Same base = add exponents!

Can I use this rule with more than two numbers?

Absolutely! For example: $2^3 \times 5^3 \times 7^3 = (2 \times 5 \times 7)^3 = 70^3$ . As long as all exponents match, combine all the bases.

How do I remember which rule to use?

Look at the exponents first! If they're the same, you can combine bases. If the bases are the same, add exponents. Different bases AND different exponents? Calculate separately.

Is it easier to calculate 12^4 or 3^4 × 4^4?

It depends on the numbers! Sometimes $12^4$ is easier, but for larger numbers, calculating $3^4 = 81$ and $4^4 = 256$ separately might be simpler.

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