Simplify the Expression: 3^4 × 3^5 Using Laws of Exponents

Q: Why do I add the exponents instead of multiplying them?

Think of it this way: $ 3^4 = 3 \times 3 \times 3 \times 3 $ and $ 3^5 = 3 \times 3 \times 3 \times 3 \times 3 $. When you multiply them together, you get 9 total factors of 3, which is $ 3^9 $!

Q: What happens if the bases are different?

If bases are different, like $ 2^3 \times 3^4 $, you cannot combine the exponents. The multiplication rule only works when the bases are identical.

Q: Do I need to calculate the final numerical value?

Not necessarily! The question asks for simplification, so $ 3^{4+5} $ is the correct simplified form. You could calculate $ 3^9 = 19683 $, but it's not required.

Q: What if I see something like 3^(2+3)?

That's already simplified using the exponent rule! $ 3^{2+3} = 3^5 $. The parentheses show you've correctly added the exponents.

Q: Can this rule work with more than two terms?

Yes! For example: $ 2^3 \times 2^4 \times 2^2 = 2^{3+4+2} = 2^9 $. Just keep adding all the exponents when the bases match.

Exponent Multiplication with Same Base

Simplify the following equation:

$3^4\times3^5=$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:07 Let's simplify this math problem together.

00:11 When we multiply numbers with the same base, let's call it A, it's like adding the exponents, N and M, together.

00:18 So, we take A, and raise it to the power of N plus M.

00:23 Let's use this rule to solve our exercise.

00:26 We'll add the exponents first, then raise the base to this new power.

00:31 And that's how we find the solution.

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Simplify the following equation:

$3^4\times3^5=$

Step-by-step solution

To solve this problem, we'll follow these steps:

Step 1: Identify the given expression and its components.
Step 2: Apply the exponent multiplication formula.
Step 3: Simplify the result.

Now, let's work through each step:
Step 1: The given expression is $3^4 \times 3^5$ . We recognize that the base is 3 and the exponents are 4 and 5.
Step 2: Apply the rule for multiplying powers with the same base: $a^m \times a^n = a^{m+n}$ . Using this formula, we add the exponents: $4 + 5$ .
Step 3: Simplify the expression: $3^{4+5} = 3^9$ .

Therefore, the simplified form of the expression is $3^{4+5}$ .

Final Answer

$3^{4+5}$

Key Points to Remember

Essential concepts to master this topic

Rule: When multiplying powers with same base, add the exponents
Technique: $3^4 \times 3^5 = 3^{4+5} = 3^9$
Check: Base stays the same, only exponents add: 3 remains 3 ✓

Common Mistakes

Avoid these frequent errors

Multiplying the exponents instead of adding them
Don't multiply exponents like $3^4 \times 3^5 = 3^{20}$ ! This gives a much larger result than correct. Always add exponents when bases are the same: $a^m \times a^n = a^{m+n}$ .

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 instead of multiplying them?

Think of it this way: $3^4 = 3 \times 3 \times 3 \times 3$ and $3^5 = 3 \times 3 \times 3 \times 3 \times 3$ . When you multiply them together, you get 9 total factors of 3, which is $3^9$ !

What happens if the bases are different?

If bases are different, like $2^3 \times 3^4$ , you cannot combine the exponents. The multiplication rule only works when the bases are identical.

Do I need to calculate the final numerical value?

Not necessarily! The question asks for simplification, so $3^{4+5}$ is the correct simplified form. You could calculate $3^9 = 19683$ , but it's not required.

What if I see something like 3^(2+3)?

That's already simplified using the exponent rule! $3^{2+3} = 3^5$ . The parentheses show you've correctly added the exponents.

Can this rule work with more than two terms?

Yes! For example: $2^3 \times 2^4 \times 2^2 = 2^{3+4+2} = 2^9$ . Just keep adding all the exponents when the bases match.

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