Solve the Power Expression: 2³ × 4³ Multiplication Problem

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

You can use this rule only when the exponents are identical. If you have $ 2^3 \times 4^3 $, both have exponent 3, so it works. But $ 2^3 \times 4^2 $ cannot be simplified this way.

Q: What's the difference between this and adding exponents?

Adding exponents applies to same bases: $ 2^3 \times 2^4 = 2^7 $. The power of product rule applies to same exponents: $ 2^3 \times 4^3 = (2 \times 4)^3 $. Different rules for different situations!

Q: Can I work backwards from the answer choices?

Absolutely! Calculate each choice to see which equals $ 2^3 \times 4^3 = 512 $. Only $ (2 \times 4)^3 = 8^3 = 512 $ will match.

Q: Does the order of multiplication matter in the base?

No! $ (2 \times 4)^3 $ equals $ (4 \times 2)^3 $ because multiplication is commutative. Both give you $ 8^3 = 512 $.

Q: What if there are more than two terms with the same exponent?

The rule extends! $ 2^3 \times 3^3 \times 5^3 = (2 \times 3 \times 5)^3 = 30^3 $. Just make sure all exponents match before combining bases.

Power of Product Rule with Same Exponents

Choose the expression that corresponds to the following:

$2^3\times4^3=$

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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 operation where each factor has the same exponent (N)

00:10 The entire multiplication can be written with the exponent (N)

00:15 We will apply this formula to our exercise

00:25 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:

$2^3\times4^3=$

Step-by-step solution

We are given the expression $2^3 \times 4^3$ and need to express it as a single term using the power of a product rule.

The power of a product rule states that any non-zero numbers $a$ and $b$ and an integer $n$ can be written as $(a \times b)^n = a^n \times b^n$ .

To apply the inverse formula, which is converting two separate powers into a product raised to a power, we look for terms that can be combined under a single exponent. Note:

Both terms $2^3$ and $4^3$ have the same exponent.
This allows us to combine them into a single expression: $(2 \times 4)^3$ .

Therefore, according to the power of a product rule applied inversely, the expression $2^3 \times 4^3$ can be rewritten as $(2 \times 4)^3$ .

Final Answer

$\left(2\times4\right)^3$

Key Points to Remember

Essential concepts to master this topic

Rule: When multiplying powers with same exponents, combine bases first
Technique: $a^n \times b^n = (a \times b)^n$ applies when exponents match
Check: Calculate both forms: $2^3 \times 4^3 = 8 \times 64 = 512$ and $(2 \times 4)^3 = 8^3 = 512$ ✓

Common Mistakes

Avoid these frequent errors

Adding exponents instead of using power of product rule
Don't write $2^3 \times 4^3 = (2 \times 4)^6$ = wrong answer! This confuses the power of product rule with the product of powers rule. Always check that exponents are the same, then combine bases: $(2 \times 4)^3$ .

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 product rule?

You can use this rule only when the exponents are identical. If you have $2^3 \times 4^3$ , both have exponent 3, so it works. But $2^3 \times 4^2$ cannot be simplified this way.

What's the difference between this and adding exponents?

Adding exponents applies to same bases: $2^3 \times 2^4 = 2^7$ . The power of product rule applies to same exponents: $2^3 \times 4^3 = (2 \times 4)^3$ . Different rules for different situations!

Can I work backwards from the answer choices?

Absolutely! Calculate each choice to see which equals $2^3 \times 4^3 = 512$ . Only $(2 \times 4)^3 = 8^3 = 512$ will match.

Does the order of multiplication matter in the base?

No! $(2 \times 4)^3$ equals $(4 \times 2)^3$ because multiplication is commutative. Both give you $8^3 = 512$ .

What if there are more than two terms with the same exponent?

The rule extends! $2^3 \times 3^3 \times 5^3 = (2 \times 3 \times 5)^3 = 30^3$ . Just make sure all exponents match before combining bases.

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