Expand the Expression: Solving (7×4×a)^5

Q: Why does the exponent apply to all factors?

The power of a product rule states that when you raise a product to a power, you must raise each factor to that same power. Think of it as multiplying the entire expression by itself 5 times!

Q: What if I have numbers and variables mixed together?

It doesn't matter! The rule works the same way whether you have numbers like 7 and 4, or variables like a. Each factor gets the exponent applied to it separately.

Q: What's the difference between this and adding exponents?

Adding exponents only applies when you have the same base being multiplied, like $ x^2 \times x^3 = x^5 $. Here we have different factors (7, 4, and a) so each gets its own exponent.

Q: Can I simplify 7×4 first before applying the exponent?

You could get $ (28a)^5 = 28^5 \times a^5 $, but it's usually better to keep factors separate. The answer $ 7^5 \times 4^5 \times a^5 $ shows your understanding of the power rule more clearly.

Power Rule with Product Expressions

Insert the corresponding expression:

$\left(7\times4\times a\right)^5=$

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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 In order to open parentheses with a multiplication operation and an outside exponent

00:07 We'll raise each factor to the power

00:11 We'll apply this formula to our exercise

00:22 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:

$\left(7\times4\times a\right)^5=$

Step-by-step solution

We begin by applying the power of a product rule to the expression $(7 \times 4 \times a)^5$ .

The power of a product rule states that $(x \times y \times z)^n = x^n \times y^n \times z^n$ .

In this case, the expression inside the parentheses is $7 \times 4 \times a$ , and it is being raised to the 5th power. We apply the exponent to each factor:

$7^5$

$4^5$

$a^5$

Therefore, the corresponding expression is $\boxed{7^5 \times 4^5 \times a^5}$ .

Final Answer

$7^5\times4^5\times a^5$

Key Points to Remember

Essential concepts to master this topic

Power of Product Rule: Distribute exponent to each factor inside parentheses
Technique: $(7 \times 4 \times a)^5 = 7^5 \times 4^5 \times a^5$
Check: Each original factor gets the same exponent (5) ✓

Common Mistakes

Avoid these frequent errors

Applying exponent to only one factor
Don't write $(7 \times 4 \times a)^5 = 7 \times 4 \times a^5$ ! This only applies the exponent to the last factor, ignoring the rule. Always distribute 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 apply to all factors?

The power of a product rule states that when you raise a product to a power, you must raise each factor to that same power. Think of it as multiplying the entire expression by itself 5 times!

What if I have numbers and variables mixed together?

It doesn't matter! The rule works the same way whether you have numbers like 7 and 4, or variables like a. Each factor gets the exponent applied to it separately.

Should I calculate 7^5 and 4^5 first?

Usually no! Unless specifically asked for a numerical answer, leave it as $7^5 \times 4^5 \times a^5$ . This shows you understand the rule and keeps the expression in its most useful form.

What's the difference between this and adding exponents?

Adding exponents only applies when you have the same base being multiplied, like $x^2 \times x^3 = x^5$ . Here we have different factors (7, 4, and a) so each gets its own exponent.

Can I simplify 7×4 first before applying the exponent?

You could get $(28a)^5 = 28^5 \times a^5$ , but it's usually better to keep factors separate. The answer $7^5 \times 4^5 \times a^5$ shows your understanding of the power rule more clearly.

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