Simplify the Expression: 9² × 9⁹ Using Exponent Properties

Q: Why do I add the exponents and not multiply them?

Think of what exponents mean! $ 9^2 $ means 9 × 9, and $ 9^9 $ means nine 9's multiplied together. When you multiply these expressions, you're combining all the factors: 2 nines + 9 nines = 11 total nines.

Q: What if the bases are different numbers?

The rule $ a^m \times a^n = a^{m+n} $ only works when the bases are identical. If you have different bases like $ 3^2 \times 5^4 $, you can't combine them using this rule.

Q: How can I remember when to add vs multiply exponents?

Multiplying same bases: Add exponentsRaising a power to a power: Multiply exponentsRemember: Multiplication of bases = Addition of exponents

Q: Can I double-check my answer without calculating the huge number?

Yes! Count the factors: $ 9^2 $ has 2 factors of 9, $ 9^9 $ has 9 factors of 9. Together that's 2 + 9 = 11 factors of 9, so the answer must be $ 9^{11} $!

Q: What's the difference between this and dividing exponents?

Great question! When dividing same bases, you subtract exponents: $ 9^9 ÷ 9^2 = 9^{9-2} = 9^7 $. But when multiplying, you always add them together.

Exponent Rules with Same Base Multiplication

Simplify the following equation:

$9^2\times9^9=$

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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, the multiplication of exponents with an equal base (A)

00:07 equals the same base raised to the sum of the exponents (N+M)

00:12 We will apply this formula to our exercise

00:15 We will then add up the exponents and raise them to this power

00:22 This is the solution

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Simplify the following equation:

$9^2\times9^9=$

Step-by-step solution

To solve this problem, we'll simplify the expression $9^2 \times 9^9$ using the multiplication of powers rule:

Step 1: Identify the base and the exponents. The base here is 9, and the exponents are 2 and 9.
Step 2: Apply the rule for multiplying powers with the same base: $a^m \times a^n = a^{m+n}$ .
Step 3: Add the exponents: $9^2 \times 9^9 = 9^{2+9} = 9^{11}$ .

The expression simplifies to $9^{11}$ .

Therefore, the simplified expression is $9^{11}$ , which matches choice 1.

Final Answer

$9^{11}$

Key Points to Remember

Essential concepts to master this topic

Rule: When multiplying same bases, add the exponents together
Technique: $9^2 \times 9^9 = 9^{2+9} = 9^{11}$
Check: Count total factors: 2 nines + 9 nines = 11 nines ✓

Common Mistakes

Avoid these frequent errors

Multiplying the exponents instead of adding them
Don't multiply 2 × 9 = 18 to get $9^{18}$ ! This creates way too many factors and gives a huge wrong answer. Always add exponents when multiplying same bases: $9^2 \times 9^9 = 9^{2+9} = 9^{11}$ .

Practice Quiz

Test your knowledge with interactive questions

$112^0=\text{?}$

FAQ

Everything you need to know about this question

Why do I add the exponents and not multiply them?

Think of what exponents mean! $9^2$ means 9 × 9, and $9^9$ means nine 9's multiplied together. When you multiply these expressions, you're combining all the factors: 2 nines + 9 nines = 11 total nines.

What if the bases are different numbers?

The rule $a^m \times a^n = a^{m+n}$ only works when the bases are identical. If you have different bases like $3^2 \times 5^4$ , you can't combine them using this rule.

How can I remember when to add vs multiply exponents?

Multiplying same bases: Add exponents
Raising a power to a power: Multiply exponents
Remember: Multiplication of bases = Addition of exponents

Can I double-check my answer without calculating the huge number?

Yes! Count the factors: $9^2$ has 2 factors of 9, $9^9$ has 9 factors of 9. Together that's 2 + 9 = 11 factors of 9, so the answer must be $9^{11}$ !

What's the difference between this and dividing exponents?

Great question! When dividing same bases, you subtract exponents: $9^9 ÷ 9^2 = 9^{9-2} = 9^7$ . But when multiplying, you always add them together.

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