Calculate (10×11×20×4×9)^8: Solving Multi-Factor Exponential Expression

Q: Why does every factor get the same exponent?

The power of a product rule states that when you raise a product to an exponent, that exponent applies to each factor equally. Think of it as multiplying the entire expression by itself 8 times!

Q: What if I forget to apply the exponent to one factor?

Your answer will be completely wrong! For example, if you miss applying the exponent to just one factor like 9, your result will be millions of times smaller than the correct answer.

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

Yes, but it's much harder! $ 10 \times 11 \times 20 \times 4 \times 9 = 79,200 $, then $ 79,200^8 $ is a massive calculation. The expanded form is easier to work with.

Q: How do I remember this rule?

Think: "Exponents distribute to everyone!" Just like sharing pizza slices equally among friends, the exponent must be shared equally among all factors in the parentheses.

Q: What if there are only two factors instead of five?

The same rule applies! Whether you have $ (a \times b)^n $ or $ (a \times b \times c \times d \times e)^n $, every factor gets the exponent: no exceptions.

Power of Products with Multiple Factors

Insert the corresponding expression:

$\left(10\times11\times20\times4\times9\right)^8=$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:11 Let's simplify this problem.

00:15 According to the laws of exponents, when a product is raised to a power, let's call it N,

00:21 it equals each factor separately raised to the power of N.

00:25 This formula works no matter how many factors there are in the product.

00:37 Let's apply this formula to our exercise.

00:41 We will break down each part and raise it to the power of N.

00:50 And that's how we solve this problem!

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Insert the corresponding expression:

$\left(10\times11\times20\times4\times9\right)^8=$

Step-by-step solution

To solve this problem, we'll apply the power of a product rule for exponents:

Given the expression:

$\left(10 \times 11 \times 20 \times 4 \times 9\right)^8$

According to the power of a product rule, which states that $(a \times b)^n = a^n \times b^n$ , we can expand this expression:

$\left(10^8 \times 11^8 \times 20^8 \times 4^8 \times 9^8\right)$

Therefore, the corresponding expanded expression is:

$10^8 \times 11^8 \times 20^8 \times 4^8 \times 9^8$

Final Answer

$10^8\times11^8\times20^8\times4^8\times9^8$

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: $(a \times b \times c)^n = a^n \times b^n \times c^n$
Check: Count factors in parentheses equals number of terms with exponents ✓

Common Mistakes

Avoid these frequent errors

Only applying the exponent to the first or last factor
Don't write $(10 \times 11 \times 20 \times 4 \times 9)^8 = 10^8 \times 11 \times 20 \times 4 \times 9$ = wrong answer! This violates the power of a product rule and gives an incorrect result. Always apply the exponent to every single 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 does every factor get the same exponent?

The power of a product rule states that when you raise a product to an exponent, that exponent applies to each factor equally. Think of it as multiplying the entire expression by itself 8 times!

What if I forget to apply the exponent to one factor?

Your answer will be completely wrong! For example, if you miss applying the exponent to just one factor like 9, your result will be millions of times smaller than the correct answer.

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

Yes, but it's much harder! $10 \times 11 \times 20 \times 4 \times 9 = 79,200$ , then $79,200^8$ is a massive calculation. The expanded form is easier to work with.

How do I remember this rule?

Think: "Exponents distribute to everyone!" Just like sharing pizza slices equally among friends, the exponent must be shared equally among all factors in the parentheses.

What if there are only two factors instead of five?

The same rule applies! Whether you have $(a \times b)^n$ or $(a \times b \times c \times d \times e)^n$ , every factor gets the exponent: no exceptions.

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