Calculate (2×5×4)^7: Evaluating a Product Raised to a Power

Q: Why does the exponent go to every number inside the parentheses?

The power of a product rule states that when you raise a product to a power, each factor gets raised to that power. Think of it as: $ (a \cdot b \cdot c)^n = a^n \cdot b^n \cdot c^n $

Q: What if I forget to apply the exponent to all factors?

You'll get the wrong answer! For example, if you write $ 2^7\times5\times4 $ instead of $ 2^7\times5^7\times4^7 $, you're missing the power rule completely.

Q: Can I calculate the product first, then raise to the power?

Yes! You could calculate $ 2\times5\times4 = 40 $, then find $ 40^7 $. But using the power rule gives you the factored form, which is often more useful.

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

Look for the parentheses! Everything inside the parentheses gets raised to the power. If it's multiplied inside, it gets the exponent.

Q: Does this work with addition inside parentheses too?

No! The power of a product rule only works with multiplication. For addition like $ (2+5)^7 $, you must calculate the sum first, then apply the exponent.

Power Rules with Multiple Factor Products

Choose the expression that corresponds to the following:

$\left(2\times5\times4\right)^7=$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Simplify the following problem

00:03 When we are presented with a multiplication operation in which each factor has 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

00:23 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:

$\left(2\times5\times4\right)^7=$

Step-by-step solution

The problem requires simplifying the expression $(2\times5\times4)^7$ using the power of a product exponent rule. According to the rule, we know that:

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

This means when we have a product raised to an exponent, each factor in the product is raised to that exponent. So let's apply this rule to the given expression:

First, identify the terms inside the parentheses: 2, 5, and 4.
Next, apply the exponent 7 to each term:
- $2^7$ – The first term 2 is raised to the power of 7.
- $5^7$ – The second term 5 is raised to the power of 7.
- $4^7$ – The third term 4 is raised to the power of 7.

Therefore, the expression $(2\times5\times4)^7$ simplifies to:

$2^7\times5^7\times4^7$

Final Answer

$2^7\times5^7\times4^7$

Key Points to Remember

Essential concepts to master this topic

Power of Product Rule: Exponent applies to every factor inside parentheses
Technique: $(2\times5\times4)^7 = 2^7\times5^7\times4^7$
Check: Each factor has same exponent as original expression ✓

Common Mistakes

Avoid these frequent errors

Applying exponent to only some factors
Don't write $2^7\times5\times4^7$ = wrong result! This breaks 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

$112^0=\text{?}$

FAQ

Everything you need to know about this question

Why does the exponent go to every number inside the parentheses?

The power of a product rule states that when you raise a product to a power, each factor gets raised to that power. Think of it as: $(a \cdot b \cdot c)^n = a^n \cdot b^n \cdot c^n$

What if I forget to apply the exponent to all factors?

You'll get the wrong answer! For example, if you write $2^7\times5\times4$ instead of $2^7\times5^7\times4^7$ , you're missing the power rule completely.

Can I calculate the product first, then raise to the power?

Yes! You could calculate $2\times5\times4 = 40$ , then find $40^7$ . But using the power rule gives you the factored form, which is often more useful.

How do I remember which factors get the exponent?

Look for the parentheses! Everything inside the parentheses gets raised to the power. If it's multiplied inside, it gets the exponent.

Does this work with addition inside parentheses too?

No! The power of a product rule only works with multiplication. For addition like $(2+5)^7$ , you must calculate the sum first, then apply the exponent.

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