Multiply Powers: Solve 8^7 × 8^8 × 9^7

Q: Why can't I just add all the exponents together?

You can only add exponents when multiplying powers with the same base. Since you have both 8 and 9 as bases, you need to use different rules for different combinations.

Q: How do I know which terms to factor together?

Look for terms with the same exponent! In this problem, $ 8^7 $ and $ 9^7 $ both have exponent 7, so they can become $ (8 \times 9)^7 $.

Q: What if none of the exponents match?

If all exponents are different, you cannot factor using exponent rules. The expression would stay as separate terms multiplied together.

Q: Can I factor the 8^8 with anything else?

$ 8^8 $ has a unique exponent of 8, so it cannot be combined with $ 8^7 $ or $ 9^7 $ using factoring. It stays separate in the final answer.

Q: How do I check if my factored form is correct?

Expand your answer back out! $ (8 \times 9)^7 \times 8^8 = 8^7 \times 9^7 \times 8^8 $. If it matches the original expression, you're right!

Exponent Rules with Common Base Factoring

Insert the corresponding expression:

$8^7\times8^8\times9^7=$

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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 According to the power laws, a product raised to a power (N)

00:06 equals a product where each factor is raised to the same power (N)

00:15 Note which factors are raised to the same power

00:18 We'll apply this formula to our exercise

00:21 Place these factors inside of parentheses raised to the power (N)

00:31 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:

$8^7\times8^8\times9^7=$

Step-by-step solution

The goal is to express the given expression $8^7 \times 8^8 \times 9^7$ using properties of exponents.

First, observe that $8^7$ and $9^7$ share a common exponent of $7$ . So, they can be factored as:

$(8 \times 9)^7$ .

This handles the product $8^7 \times 9^7$ . Now, include $8^8$ which is not part of the factoring:

$(8 \times 9)^7 \times 8^8$ .

This resulting expression matches the provided possible choice.

Therefore, the rewritten expression is $\left(8 \times 9\right)^7 \times 8^8$ .

Final Answer

$\left(8\times9\right)^7\times8^8$

Key Points to Remember

Essential concepts to master this topic

Same Base Rule: Add exponents when multiplying powers with identical bases
Common Exponent Rule: Factor out shared exponents like $8^7 \times 9^7 = (8 \times 9)^7$
Check: Expand your factored form to verify it equals the original expression ✓

Common Mistakes

Avoid these frequent errors

Trying to combine all terms into one exponent
Don't write $8^7 \times 8^8 \times 9^7 = (8 \times 8 \times 9)^{15}$ ! This ignores different bases and exponents. You can't combine terms with different bases or different exponents into a single power. Always group terms with matching exponents first, then handle remaining terms separately.

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 all the exponents together?

You can only add exponents when multiplying powers with the same base. Since you have both 8 and 9 as bases, you need to use different rules for different combinations.

How do I know which terms to factor together?

Look for terms with the same exponent! In this problem, $8^7$ and $9^7$ both have exponent 7, so they can become $(8 \times 9)^7$ .

What if none of the exponents match?

If all exponents are different, you cannot factor using exponent rules. The expression would stay as separate terms multiplied together.

Can I factor the 8^8 with anything else?

$8^8$ has a unique exponent of 8, so it cannot be combined with $8^7$ or $9^7$ using factoring. It stays separate in the final answer.

How do I check if my factored form is correct?

Expand your answer back out! $(8 \times 9)^7 \times 8^8 = 8^7 \times 9^7 \times 8^8$ . If it matches the original expression, you're right!

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