Simplify the Expression: 9×9⁹ Power and Multiplication Problem

Q: Why do I need to write 9 as 9¹?

Writing 9 as $ 9^1 $ makes the exponent rule visible and applicable. Any number without an exponent actually has an invisible exponent of 1, so showing it helps you use the rule $ a^m \times a^n = a^{m+n} $.

Q: What's the difference between 9⁹ and 9¹⁰?

$ 9^9 $ means 9 multiplied by itself 9 times, while $ 9^{10} $ means 9 multiplied by itself 10 times. So $ 9^{10} = 9^9 \times 9 $ - that's exactly our original problem!

Q: Can I just calculate 9 × 9⁹ by finding 9⁹ first?

You could, but $ 9^9 = 387,420,489 $, so you'd multiply by a huge number! Using exponent rules gives you the simplified form $ 9^{10} $ without messy calculations.

Q: Do I add or multiply the exponents?

When multiplying same bases, you add the exponents: $ a^m \times a^n = a^{m+n} $. When raising a power to a power, you multiply exponents: $ (a^m)^n = a^{m \times n} $.

Q: Why is 9^(1+9) the correct answer choice?

The answer choice $ 9^{1+9} $ shows the process step before final simplification. It demonstrates that you correctly applied the exponent rule by adding 1 + 9, even though it simplifies further to $ 9^{10} $.

Exponent Rules with Base Multiplication

Simplify the following equation:

$9\times9^9=$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:06 Let's simplify this expression together.

00:09 Here's a helpful rule to remember: when we multiply terms with the same base A, we keep the base and add the exponents.

00:17 So, A to the power of N times A to the power of M becomes A to the power of N plus M.

00:23 Remember another useful rule: any number raised to the power of 1 is just the number itself.

00:29 Now, let's apply these rules to our problem.

00:33 We'll add the exponents together and keep our base, giving us our simplified answer.

00:39 And there we have it! That's our simplified solution.

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Simplify the following equation:

$9\times9^9=$

Step-by-step solution

To solve this problem, let's apply the multiplication of powers rule:

Step 1: Identify expression as $9 \times 9^9$ .
Step 2: Note that $9$ can be expressed as $9^1$ .
Step 3: Apply the exponent rule: $a^m \times a^n = a^{m+n}$ .

Now, we'll work through the calculation step-by-step:

Step 1: Rewrite $9$ as $9^1$ . Thus, our expression becomes $9^1 \times 9^9$ .

Step 2: Use the exponent rule to combine: $9^1 \times 9^9 = 9^{1+9}$ .

Step 3: Simplify the exponent by adding: $9^{1+9} = 9^{10}$ .

Therefore, the simplified form of the expression is $9^{10}$ .

In terms of the answer choices, the correct answer is

$9^{1+9}$

Final Answer

$9^{1+9}$

Key Points to Remember

Essential concepts to master this topic

Rule: When multiplying same bases, add the exponents together
Technique: Rewrite 9 as $9^1$ so $9 \times 9^9 = 9^1 \times 9^9$
Check: Verify $9^{1+9} = 9^{10}$ by confirming exponent addition ✓

Common Mistakes

Avoid these frequent errors

Adding the bases instead of the exponents
Don't calculate 9 + 9⁹ or think 9 × 9⁹ = 18⁹! This completely ignores exponent rules and gives astronomically wrong answers. Always rewrite the base as a power first, then add only the exponents when multiplying same bases.

Practice Quiz

Test your knowledge with interactive questions

$112^0=\text{?}$

FAQ

Everything you need to know about this question

Why do I need to write 9 as 9¹?

Writing 9 as $9^1$ makes the exponent rule visible and applicable. Any number without an exponent actually has an invisible exponent of 1, so showing it helps you use the rule $a^m \times a^n = a^{m+n}$ .

What's the difference between 9⁹ and 9¹⁰?

$9^9$ means 9 multiplied by itself 9 times, while $9^{10}$ means 9 multiplied by itself 10 times. So $9^{10} = 9^9 \times 9$ - that's exactly our original problem!

Can I just calculate 9 × 9⁹ by finding 9⁹ first?

You could, but $9^9 = 387,420,489$ , so you'd multiply by a huge number! Using exponent rules gives you the simplified form $9^{10}$ without messy calculations.

Do I add or multiply the exponents?

When multiplying same bases, you add the exponents: $a^m \times a^n = a^{m+n}$ . When raising a power to a power, you multiply exponents: $(a^m)^n = a^{m \times n}$ .

Why is 9^(1+9) the correct answer choice?

The answer choice $9^{1+9}$ shows the process step before final simplification. It demonstrates that you correctly applied the exponent rule by adding 1 + 9, even though it simplifies further to $9^{10}$ .

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