Evaluate (5×2×6×3)^7: Product and Power Expression

Q: Why can't I just multiply the numbers first and then raise to the 7th power?

You absolutely can do that! $ 5 \times 2 \times 6 \times 3 = 180 $, so $ 180^7 $ gives the same result. However, the question asks for the expression form, not the numerical calculation.

Q: How do I remember to apply the exponent to every factor?

Think of the exponent as a multiplier that affects everything inside the parentheses equally. Just like distributing multiplication, you must distribute the power to each factor: no exceptions!

Q: What if some factors are the same number?

Each factor gets its own power regardless! For example, $ (2 \times 2 \times 3)^4 = 2^4 \times 2^4 \times 3^4 $. You can combine same bases later using $ 2^4 \times 2^4 = 2^8 $.

Q: Does the order of factors matter when applying the power rule?

No! Multiplication is commutative, so $ 5^7 \times 2^7 \times 6^7 \times 3^7 $ equals $ 2^7 \times 3^7 \times 5^7 \times 6^7 $. The important thing is that every factor gets the exponent.

Q: Can I group some factors together before applying the power?

Yes! You could write $ (5 \times 3)^7 \times (2 \times 6)^7 = 15^7 \times 12^7 $, but the question specifically wants each individual factor raised to the 7th power.

Power of Products with Multiple Factors

Insert the corresponding expression:

$\left(5\times2\times6\times3\right)^7=$

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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 According to the laws of exponents, a product raised to a power (N)

00:10 equals a product where each factor is raised to the same power (N)

00:15 This formula is valid regardless of how many factors are in the product

00:21 We will apply this formula to our exercise

00:26 We will break down the product into each factor separately raised to the power (N)

00:34 This is the solution

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Insert the corresponding expression:

$\left(5\times2\times6\times3\right)^7=$

Step-by-step solution

To solve this problem, we'll convert the given product inside the parentheses, $\left(5 \times 2 \times 6 \times 3\right)^7$ , into an expression applying the power of a product rule.

First, recognize that we can apply the rule of exponents as follows:

The power of a product rule states $(a \times b)^n = a^n \times b^n$ .

Applying this to our variables, we have:

$\left(5 \times 2 \times 6 \times 3\right)^7 = (5)^7 \times (2)^7 \times (6)^7 \times (3)^7$

Therefore, the expression evaluates to:

$5^7 \times 2^7 \times 6^7 \times 3^7$

Final Answer

$5^7\times2^7\times6^7\times3^7$

Key Points to Remember

Essential concepts to master this topic

Power Rule: Distribute exponent to each factor in the product
Technique: $(a \times b)^n = a^n \times b^n$ for all factors
Check: Count factors inside and outside: 4 factors become 4 separate powers ✓

Common Mistakes

Avoid these frequent errors

Only applying the exponent to some factors
Don't raise just some factors to the 7th power like $5^7 \times 2 \times 6^7 \times 3$ = wrong distribution! The power rule requires the exponent to apply to every single factor inside the parentheses. Always distribute the exponent to each and every factor: $5^7 \times 2^7 \times 6^7 \times 3^7$ .

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 the numbers first and then raise to the 7th power?

You absolutely can do that! $5 \times 2 \times 6 \times 3 = 180$ , so $180^7$ gives the same result. However, the question asks for the expression form, not the numerical calculation.

How do I remember to apply the exponent to every factor?

Think of the exponent as a multiplier that affects everything inside the parentheses equally. Just like distributing multiplication, you must distribute the power to each factor: no exceptions!

What if some factors are the same number?

Each factor gets its own power regardless! For example, $(2 \times 2 \times 3)^4 = 2^4 \times 2^4 \times 3^4$ . You can combine same bases later using $2^4 \times 2^4 = 2^8$ .

Does the order of factors matter when applying the power rule?

No! Multiplication is commutative, so $5^7 \times 2^7 \times 6^7 \times 3^7$ equals $2^7 \times 3^7 \times 5^7 \times 6^7$ . The important thing is that every factor gets the exponent.

Can I group some factors together before applying the power?

Yes! You could write $(5 \times 3)^7 \times (2 \times 6)^7 = 15^7 \times 12^7$ , but the question specifically wants each individual factor raised to the 7th power.

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