Calculate (2×6×8)⁴: Evaluating a Power of Multiple Numbers

Q: Why does the exponent go on every factor?

The power of a product rule says $ (a \cdot b \cdot c)^n = a^n \cdot b^n \cdot c^n $. Think of it like this: you're multiplying the entire product by itself 4 times, so each factor appears 4 times total!

Q: What if I just calculate 2×6×8 first, then raise to the 4th power?

That works too! $ 2 \times 6 \times 8 = 96 $, so $ (2 \times 6 \times 8)^4 = 96^4 $. But the question asks for the equivalent expression, not the numerical answer.

Q: How do I remember which factors get the exponent?

Every factor inside the parentheses gets the exponent! If it's being multiplied inside the ( ), it gets raised to the power. Nothing outside the parentheses is affected.

Q: Does this work with addition inside parentheses too?

No! The power rule only works with multiplication. For addition like $ (2 + 6)^4 $, you must add first, then raise to the power. You cannot distribute exponents over addition.

Q: What if there are different exponents on the factors already?

Use the power of a power rule! For example, $ (2^3 \times 6^2)^4 = 2^{3 \times 4} \times 6^{2 \times 4} = 2^{12} \times 6^8 $. Multiply the exponents together.

Power Rules with Multiple Factor Products

Choose the expression that corresponds to the following:

$\left(2\times6\times8\right)^4=$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Simplify the following problem

00:04 When we are presented with a multiplication operation where all the factors have the same exponent (N)

00:07 Each factor can be raised to the power (N)

00:13 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:

$\left(2\times6\times8\right)^4=$

Step-by-step solution

To solve the question, we need to apply the power of a product rule from exponents. This rule states that when a product is raised to an exponent, we can apply the exponent to each factor within the product individually. Mathematically, the rule is expressed as:

$(a \cdot b \cdot c)^n = a^n \cdot b^n \cdot c^n$ .

Now, we identify the components in the given expression:

The expression inside the parentheses is $2 \times 6 \times 8$ .
The exponent applied to this product is $4$ .

Applying the exponent to each factor gives us:

Apply the exponent 4 to the factor 2: $2^4$ .
Apply the exponent 4 to the factor 6: $6^4$ .
Apply the exponent 4 to the factor 8: $8^4$ .

Therefore, the expression $(2 \times 6 \times 8)^4$ is transformed into:

$2^4 \times 6^4 \times 8^4$ .

This matches the correct answer provided: $2^4 \times 6^4 \times 8^4$ .

Final Answer

$2^4\times6^4\times8^4$

Key Points to Remember

Essential concepts to master this topic

Power Rule: When raising a product to a power, apply exponent to each factor
Technique: $(2 \times 6 \times 8)^4 = 2^4 \times 6^4 \times 8^4$
Check: Each factor inside parentheses gets the same exponent 4 ✓

Common Mistakes

Avoid these frequent errors

Applying exponent to only some factors
Don't write $2^4 \times 6 \times 8^4$ or similar = wrong result! This violates the power rule and gives an incorrect expression. Always apply the exponent to every single factor inside the parentheses.

Practice Quiz

Test your knowledge with interactive questions

$$ (4^2)^3+(g^3)^4= $$

FAQ

Everything you need to know about this question

Why does the exponent go on every factor?

The power of a product rule says $(a \cdot b \cdot c)^n = a^n \cdot b^n \cdot c^n$ . Think of it like this: you're multiplying the entire product by itself 4 times, so each factor appears 4 times total!

What if I just calculate 2×6×8 first, then raise to the 4th power?

That works too! $2 \times 6 \times 8 = 96$ , so $(2 \times 6 \times 8)^4 = 96^4$ . But the question asks for the equivalent expression, not the numerical answer.

How do I remember which factors get the exponent?

Every factor inside the parentheses gets the exponent! If it's being multiplied inside the ( ), it gets raised to the power. Nothing outside the parentheses is affected.

Does this work with addition inside parentheses too?

No! The power rule only works with multiplication. For addition like $(2 + 6)^4$ , you must add first, then raise to the power. You cannot distribute exponents over addition.

What if there are different exponents on the factors already?

Use the power of a power rule! For example, $(2^3 \times 6^2)^4 = 2^{3 \times 4} \times 6^{2 \times 4} = 2^{12} \times 6^8$ . Multiply the exponents together.

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