Calculate the Product: Solving 8^5 × 9^5

Q: Why can't I just add the exponents like in other problems?

You can only add exponents when the bases are the same! For example: $ 8^3 \times 8^2 = 8^5 $. But when bases are different like 8 and 9, use the product rule instead.

Q: How do I know when to use the product rule?

Use the product rule $ a^n \times b^n = (a \times b)^n $ when you see different bases with the same exponent. If the bases were the same, you'd add exponents instead.

Q: Are (8×9)⁵ and (9×8)⁵ really the same?

Yes! Multiplication is commutative, meaning order doesn't matter. Both equal $ 72^5 $ because 8 × 9 = 9 × 8 = 72.

Q: What if the exponents were different?

If exponents don't match, you cannot use the product rule! For example, $ 8^3 \times 9^5 $ stays as is - no simplification possible with basic rules.

Q: Should I always calculate 72⁵ to get the final number?

Not necessarily! The expression $ 72^5 $ is already simplified. Computing the actual value (over 19 million) isn't always required unless specifically asked.

Exponent Rules with Product Properties

Choose the expression that corresponds to the following:

$8^5\times9^5=$

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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 When we are presented with a product where each factor has the same exponent (N)

00:06 The entire product can be written with the exponent (N)

00:13 We will apply this formula to our exercise

00:17 In multiplication, the order of factors doesn't matter, therefore the expressions are equal

00:27 We will apply this formula to our exercise

00:39 This is one potential solution, let's proceed to review another possible solution

00:52 We'll again use the formula for the power of a product

00:58 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:

$8^5\times9^5=$

Step-by-step solution

The problem given is to simplify the expression $8^5 \times 9^5$ . This problem can be solved by applying the power of a product rule exponent rule.

Step-by-step solution:

According to the power of a product rule, when two expressions with the same exponent are multiplied, the product can be written as a single power expression:
$a^n \times b^n = (a \times b)^n$
Using this rule, we can rewrite the given expression $8^5 \times 9^5$ as a single power:
$(8 \times 9)^5$
Therefore, the expression simplifies to:
$(72)^5$
We have represented $8^5 \times 9^5$ as $(72)^5$ , which is its corresponding expression according to the power of a product rule.

By recognizing that $8^5 \times 9^5$ can be expressed as a single power of $72$ , we confirm that the problem demonstrates the application of the power of a product rule. All of the expressions are mathematically the same and therefore the correct answer is (d) "All answers are correct".

Final Answer

All answers are correct.

Key Points to Remember

Essential concepts to master this topic

Rule: When bases differ but exponents match: $a^n \times b^n = (a \times b)^n$
Technique: Combine bases first: $8^5 \times 9^5 = (8 \times 9)^5 = 72^5$
Check: All equivalent forms are mathematically identical: $(8 \times 9)^5 = (9 \times 8)^5 = 72^5$ ✓

Common Mistakes

Avoid these frequent errors

Adding exponents when multiplying different bases
Don't write $8^5 \times 9^5 = 72^{10}$ by adding 5 + 5 = wrong result! This only works when bases are identical. Always use the product rule: combine bases first, keep the same exponent.

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 add the exponents like in other problems?

You can only add exponents when the bases are the same! For example: $8^3 \times 8^2 = 8^5$ . But when bases are different like 8 and 9, use the product rule instead.

How do I know when to use the product rule?

Use the product rule $a^n \times b^n = (a \times b)^n$ when you see different bases with the same exponent. If the bases were the same, you'd add exponents instead.

Are (8×9)⁵ and (9×8)⁵ really the same?

Yes! Multiplication is commutative, meaning order doesn't matter. Both equal $72^5$ because 8 × 9 = 9 × 8 = 72.

What if the exponents were different?

If exponents don't match, you cannot use the product rule! For example, $8^3 \times 9^5$ stays as is - no simplification possible with basic rules.

Should I always calculate 72⁵ to get the final number?

Not necessarily! The expression $72^5$ is already simplified. Computing the actual value (over 19 million) isn't always required unless specifically asked.

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