Multiply (3^4) × (3^2): Solving Powers with Same Base

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

Think of what exponents mean! $ 3^4 = 3 \times 3 \times 3 \times 3 $ and $ 3^2 = 3 \times 3 $. When you multiply them, you get six 3's multiplied together, which equals $ 3^6 $.

Q: What if the bases are different, like $ 3^4 \times 2^2 $ ?

You cannot use this rule when bases are different! You'd have to calculate each power separately: $ 3^4 = 81 $ and $ 2^2 = 4 $, then multiply: $ 81 \times 4 = 324 $.

Q: How can I remember this rule?

Think of it as "Same base, add the powers!" The formula is $ a^m \times a^n = a^{m+n} $. Practice with simple examples like $ 2^3 \times 2^2 = 2^5 $ to build confidence.

Q: Can I check my answer without calculating the big numbers?

Yes! Use the reverse rule: $ 3^6 = 3^4 \times 3^2 $. If your answer follows the pattern correctly and the exponents add up right, you're on the right track!

Q: What if one of the exponents is negative?

The same rule applies! For example: $ 3^4 \times 3^{-2} = 3^{4+(-2)} = 3^2 $. Just add the exponents like normal, even with negative numbers.

Exponent Rules with Same Base Multiplication

Solve the following problem:

$\left(3^4\right)\times\left(3^2\right)=$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Let's solve it

00:03 According to the laws of exponents, when multiplying powers with the same base (A)

00:09 We get the same base (A) raised to the sum of the exponents (M+N)

00:13 Let's use this formula in our exercise

00:23 Let's compare terms according to the formula and simplify

01:18 Let's keep the base

01:29 Let's add the exponents

02:00 Let's calculate the sum of exponents

02:21 And this is the solution to the question

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Solve the following problem:

$\left(3^4\right)\times\left(3^2\right)=$

Step-by-step solution

In order to solve this problem, we'll follow these steps:

Step 1: Identify the base and exponents
Step 2: Use the formula for multiplying powers with the same base
Step 3: Simplify the expression by applying the relevant exponent rule

Now, let's work through each step:

Step 1: The given expression is $(3^4) \times (3^2)$ . Here, the base is 3, and the exponents are 4 and 2.

Step 2: Apply the exponent rule, which states that when multiplying powers with the same base, we add the exponents:
$a^m \times a^n = a^{m+n}$

Step 3: Using the rule identified in Step 2, we add the exponents 4 and 2:
$3^4 \times 3^2 = 3^{4+2} = 3^6$

Therefore, the simplified form of the expression is $3^6$ .

Final Answer

$3^6$

Key Points to Remember

Essential concepts to master this topic

Rule: When multiplying same bases, add the exponents together
Technique: $3^4 \times 3^2 = 3^{4+2} = 3^6$
Check: Verify by calculating: $3^4 = 81$ and $3^2 = 9$ , so $81 \times 9 = 729 = 3^6$ ✓

Common Mistakes

Avoid these frequent errors

Multiplying the exponents instead of adding them
Don't multiply the exponents: $3^4 \times 3^2 ≠ 3^{4×2} = 3^8$ ! This gives 6561 instead of the correct answer 729. Always add the exponents when multiplying powers with the same base.

Practice Quiz

Test your knowledge with interactive questions

$112^0=\text{?}$

FAQ

Everything you need to know about this question

Why do we add the exponents instead of multiplying them?

Think of what exponents mean! $3^4 = 3 \times 3 \times 3 \times 3$ and $3^2 = 3 \times 3$ . When you multiply them, you get six 3's multiplied together, which equals $3^6$ .

What if the bases are different, like $3^4 \times 2^2$ ?

You cannot use this rule when bases are different! You'd have to calculate each power separately: $3^4 = 81$ and $2^2 = 4$ , then multiply: $81 \times 4 = 324$ .

How can I remember this rule?

Think of it as "Same base, add the powers!" The formula is $a^m \times a^n = a^{m+n}$ . Practice with simple examples like $2^3 \times 2^2 = 2^5$ to build confidence.

Can I check my answer without calculating the big numbers?

Yes! Use the reverse rule: $3^6 = 3^4 \times 3^2$ . If your answer follows the pattern correctly and the exponents add up right, you're on the right track!

What if one of the exponents is negative?

The same rule applies! For example: $3^4 \times 3^{-2} = 3^{4+(-2)} = 3^2$ . Just add the exponents like normal, even with negative numbers.

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Multiply (3^4) × (3^2): Solving Powers with Same Base

Exponent Rules with Same Base Multiplication

❤️ Continue Your Math Journey!

Step-by-step video solution

Step-by-step written solution

Understand the problem

Step-by-step solution

Final Answer

Key Points to Remember

Common Mistakes

Practice Quiz

FAQ

Why do we add the exponents instead of multiplying them?

What if the bases are different, like $3^4 \times 2^2$ ?

How can I remember this rule?

Can I check my answer without calculating the big numbers?

What if one of the exponents is negative?

🌟 Unlock Your Math Potential

More Questions

Applying Combined Exponents Rules

Simplify the Expression: x⁴·x³/x⁵·x² Using Exponent Rules

Simplify the Expression: 3⁵ ÷ 3² Using Exponent Rules

Complete the Expression: Writing 10^3x in Standard Form

Complete the Expression: Finding (15)^xy Power with Multiple Variables

Multiply (3^4) × (3^2): Solving Powers with Same Base

Exponent Rules with Same Base Multiplication

❤️ Continue Your Math Journey!

Step-by-step video solution

Step-by-step written solution

Understand the problem

Step-by-step solution

Final Answer

Key Points to Remember

Common Mistakes

Practice Quiz

FAQ

Why do we add the exponents instead of multiplying them?

What if the bases are different, like 34×22 3^4 \times 2^2 34×22?

How can I remember this rule?

Can I check my answer without calculating the big numbers?

What if one of the exponents is negative?

🌟 Unlock Your Math Potential

More Questions

Applying Combined Exponents Rules

Simplify the Expression: x⁴·x³/x⁵·x² Using Exponent Rules

Simplify the Expression: 3⁵ ÷ 3² Using Exponent Rules

Complete the Expression: Writing 10^3x in Standard Form

Complete the Expression: Finding (15)^xy Power with Multiple Variables

What if the bases are different, like $3^4 \times 2^2$ ?