Solve the Power Expression: 10^12 × 9^6 × 9^12

Q: Why can't I combine all three terms into one expression?

The property $ a^m \times b^m = (a \times b)^m $ only works when the exponents are identical. Since we have $ 9^6 $ and $ 9^{12} $ with different exponents, they can't all be combined together.

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

Look for identical exponents! In this problem, both $ 10^{12} $ and $ 9^{12} $ have exponent 12, so they can be combined as $ (10 \times 9)^{12} $.

Q: What if the bases were the same instead of the exponents?

If bases were the same, you'd use $ a^m \times a^n = a^{m+n} $. For example: $ 9^6 \times 9^{12} = 9^{18} $. But here we have different bases with same exponents.

Q: Is there a specific order I should follow?

Yes! First: Identify terms with identical exponents. Second: Apply $ a^m \times b^m = (a \times b)^m $. Third: Keep any remaining terms separate.

Q: Can I simplify (10 × 9)^12 × 9^6 further?

You could calculate $ 10 \times 9 = 90 $ to get $ 90^{12} \times 9^6 $, but the form $ (10 \times 9)^{12} \times 9^6 $ clearly shows the factoring work and is the expected answer.

Exponent Properties with Same Base Combinations

Insert the corresponding expression:

$10^{12}\times9^6\times9^{12}=$

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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 laws of exponents, a product that is raised to the power (N)

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

00:16 Note which factors are raised to the same power

00:20 We'll apply this formula to our exercise

00:23 We'll place these factors inside of parentheses raised to the power (N)

00:30 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:

$10^{12}\times9^6\times9^{12}=$

Step-by-step solution

Given $10^{12}$ and $9^{12}$ , apply the property $a^m \times b^m = (a \times b)^m$ , to rewrite part of the expression as:

$10^{12} \times 9^{12} = (10 \times 9)^{12}$ .

The expression now becomes:

$(10 \times 9)^{12} \times 9^6$ .

Therefore, the expression $10^{12} \times 9^6 \times 9^{12}$ simplifies to $(10 \times 9)^{12} \times 9^6$ .

Final Answer

$\left(10\times9\right)^{12}\times9^6$

Key Points to Remember

Essential concepts to master this topic

Property: When bases are same, $a^m \times b^m = (a \times b)^m$
Technique: Group same exponents: $10^{12} \times 9^{12} = (10 \times 9)^{12}$
Check: Final form should have fewer terms than original expression ✓

Common Mistakes

Avoid these frequent errors

Combining all terms incorrectly into one power
Don't write $(10 \times 9 \times 9)^{18}$ by adding all exponents = wrong grouping! This ignores that 9^6 and 9^12 have different exponents. Always identify terms with identical exponents first, then apply the property $a^m \times b^m = (a \times b)^m$ .

Practice Quiz

Test your knowledge with interactive questions

$112^0=\text{?}$

FAQ

Everything you need to know about this question

Why can't I combine all three terms into one expression?

The property $a^m \times b^m = (a \times b)^m$ only works when the exponents are identical. Since we have $9^6$ and $9^{12}$ with different exponents, they can't all be combined together.

How do I know which terms to group together?

Look for identical exponents! In this problem, both $10^{12}$ and $9^{12}$ have exponent 12, so they can be combined as $(10 \times 9)^{12}$ .

What if the bases were the same instead of the exponents?

If bases were the same, you'd use $a^m \times a^n = a^{m+n}$ . For example: $9^6 \times 9^{12} = 9^{18}$ . But here we have different bases with same exponents.

Is there a specific order I should follow?

Yes! First: Identify terms with identical exponents. Second: Apply $a^m \times b^m = (a \times b)^m$ . Third: Keep any remaining terms separate.

Can I simplify (10 × 9)^12 × 9^6 further?

You could calculate $10 \times 9 = 90$ to get $90^{12} \times 9^6$ , but the form $(10 \times 9)^{12} \times 9^6$ clearly shows the factoring work and is the expected answer.

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