Calculate (9×7)⁴: Fourth Power of Product Exercise

Q: What's the difference between (a×b)ⁿ and aⁿ×bⁿ?

They're exactly the same! The power of a product rule states that $ (a \times b)^n = a^n \times b^n $. This rule lets you distribute exponents across multiplication.

Q: Does this work with more than two factors?

Yes! For example, $ (2 \times 3 \times 5)^2 = 2^2 \times 3^2 \times 5^2 $. The exponent distributes to every single factor inside the parentheses.

Q: What if the exponent was on the outside of just one factor?

That would be different! $ 9 \times 7^4 $ means only 7 gets raised to the 4th power, while 9 stays as is. Always check where the parentheses are placed.

Q: How do I remember this rule?

Think: "The exponent is generous - it shares with everyone inside the parentheses!" Every factor gets the same exponent when you distribute it.

Power of Products with Exponent Distribution

Choose the expression that represents the following:

$\left(9\times7\right)^4=$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Simplify the following expression

00:03 In order to open parentheses with an exponent over multiplication

00:06 Raise each factor to the power

00:13 We will use this formula in our exercise

00:17 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 represents the following:

$\left(9\times7\right)^4=$

Step-by-step solution

To solve the problem, we need to apply the power of a product rule for exponents, which states that if you have a product raised to an exponent, you can apply the exponent to each factor in the product individually.

The general form of this rule is:

$(a \times b)^n = a^n \times b^n$

According to this formula, when we have the expression $(9 \times 7)^4$ we apply the exponent 4 to each factor within the parentheses.

This process results in:

Raising 9 to the power of 4: $9^4$
Raising 7 to the power of 4: $7^4$

Therefore, the expression simplifies to:

$9^4 \times 7^4$

Final Answer

$9^4\times7^4$

Key Points to Remember

Essential concepts to master this topic

Rule: Power of a product distributes to each factor separately
Technique: $(9 \times 7)^4 = 9^4 \times 7^4$ by applying exponent to each
Check: Both equal 2,541,865 when calculated separately ✓

Common Mistakes

Avoid these frequent errors

Adding exponents instead of distributing them
Don't write $(9 \times 7)^4 = 9 \times 7^4$ = 21,609! This applies the exponent to only one factor. Always distribute the exponent to every 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 can't I just calculate 9×7=63 first, then find 63⁴?

You absolutely can! Both methods give the same answer. The question asks for the equivalent expression using the power rule, which shows $9^4 \times 7^4$ without calculating the product first.

What's the difference between (a×b)ⁿ and aⁿ×bⁿ?

They're exactly the same! The power of a product rule states that $(a \times b)^n = a^n \times b^n$ . This rule lets you distribute exponents across multiplication.

Does this work with more than two factors?

Yes! For example, $(2 \times 3 \times 5)^2 = 2^2 \times 3^2 \times 5^2$ . The exponent distributes to every single factor inside the parentheses.

What if the exponent was on the outside of just one factor?

That would be different! $9 \times 7^4$ means only 7 gets raised to the 4th power, while 9 stays as is. Always check where the parentheses are placed.

How do I remember this rule?

Think: "The exponent is generous - it shares with everyone inside the parentheses!" Every factor gets the same exponent when you distribute it.

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