Calculate (8×7×3)^8: Evaluating a Product Raised to a Power

Q: Why are all three answer choices correct?

All three options show the same multiplication just in different orders! Since multiplication is commutative, $ 8^8 × 7^8 × 3^8 $ equals $ 7^8 × 3^8 × 8^8 $ and any other arrangement.

Q: Can I calculate the numbers inside first, then raise to the 8th power?

Yes! You could compute $ 8×7×3 = 168 $, then find $ 168^8 $. However, this creates much larger numbers and is harder to work with than keeping factors separate.

Q: What's the power of a product rule exactly?

The rule states: $ (a×b×c)^n = a^n × b^n × c^n $. When you raise a product to a power, distribute that exponent to every factor inside the parentheses.

Q: Does this work with more than three factors?

Absolutely! The power of a product rule works for any number of factors. For example: $ (2×3×4×5)^3 = 2^3 × 3^3 × 4^3 × 5^3 $.

Q: What if I have variables instead of numbers?

The same rule applies! $ (xy)^3 = x^3y^3 $ and $ (2ab)^4 = 2^4a^4b^4 = 16a^4b^4 $. Just distribute the exponent to each factor.

Power of Products with Multiple Factors

Choose the expression that corresponds to the following:

$\left(8\times7\times3\right)^8=$

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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)

00:08 Each term can be raised to the power (N)

00:11 We will apply this formula to our exercise

00:19 In multiplication, the order of factors doesn't matter, therefore the expressions are equal

00:39 We will apply this formula to our exercise and change the order of factors

00:55 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:

$\left(8\times7\times3\right)^8=$

Step-by-step solution

To solve the expression $\left(8\times7\times3\right)^8$ , we can use the power of a product rule. This rule states that when raising a product to an exponent, you can apply the exponent to each factor within the parentheses.

So, according to the rule:

$\left(8\times7\times3\right)^8 = 8^8 \times 7^8 \times 3^8$

Each of the factors: 8, 7, and 3 is independently raised to the power of 8.

This approach allows us to separate the original power into the power of each individual factor, making the expression equivalent to multiplying each of these results together.

Therefore, the corresponding expression that equals $\left(8\times7\times3\right)^8$ is:

$8^8 \times 7^8 \times 3^8$
Each factor separately raised to the 8th power, then multiplied together.

All answers similar to this transformation are correct, as they apply the correct exponent rules.

Final Answer

All answers are correct.

Key Points to Remember

Essential concepts to master this topic

Rule: Apply exponent to each factor inside parentheses separately
Technique: $(8×7×3)^8 = 8^8 × 7^8 × 3^8$
Check: All three answer choices show same multiplication in different orders ✓

Common Mistakes

Avoid these frequent errors

Adding exponents instead of applying to each factor
Don't write $(8×7×3)^8 = 8^{8+7+3}$ = wrong result! This confuses multiplication with exponentiation rules. Always apply the exponent to each individual 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 are all three answer choices correct?

All three options show the same multiplication just in different orders! Since multiplication is commutative, $8^8 × 7^8 × 3^8$ equals $7^8 × 3^8 × 8^8$ and any other arrangement.

Can I calculate the numbers inside first, then raise to the 8th power?

Yes! You could compute $8×7×3 = 168$ , then find $168^8$ . However, this creates much larger numbers and is harder to work with than keeping factors separate.

What's the power of a product rule exactly?

The rule states: $(a×b×c)^n = a^n × b^n × c^n$ . When you raise a product to a power, distribute that exponent to every factor inside the parentheses.

Does this work with more than three factors?

Absolutely! The power of a product rule works for any number of factors. For example: $(2×3×4×5)^3 = 2^3 × 3^3 × 4^3 × 5^3$ .

What if I have variables instead of numbers?

The same rule applies! $(xy)^3 = x^3y^3$ and $(2ab)^4 = 2^4a^4b^4 = 16a^4b^4$ . Just distribute the exponent to each factor.

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