Solve 3^4 × 3^x: Multiplying Powers with Same Base

Q: What if the bases were different numbers?

If the bases are different (like $ 2^4 \times 3^x $), you cannot combine them using this rule. The multiplication rule for exponents only works when the bases are identical.

Q: Does this rule work with negative exponents too?

Yes! The rule $ a^m \times a^n = a^{m+n} $ works for all exponents - positive, negative, fractions, or variables. Just add them normally.

Q: How is this different from raising a power to a power?

Great question! $ 3^4 \times 3^x = 3^{4+x} $ (add exponents) is different from $ (3^4)^x = 3^{4 \cdot x} $ (multiply exponents). Watch for multiplication vs. parentheses!

Q: Can I use this rule with more than two terms?

Absolutely! For example: $ 3^2 \times 3^4 \times 3^x = 3^{2+4+x} = 3^{6+x} $. Just keep adding all the exponents when the bases are the same.

Exponent Laws with Variable Powers

$3^4\times3^x=$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Simplify the following problem

00:04 According to the laws of exponents, the multiplication of exponents with equal bases (A)

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

00:10 We'll apply this formula to our exercise

00:13 We'll maintain the base and add the exponents together

00:16 This is the solution

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

$3^4\times3^x=$

Step-by-step solution

To solve this problem, we'll apply the exponent rule for multiplying powers with the same base.

Step 1: Identify the base. The base for both terms is 3.
Step 2: Apply the multiplication of powers rule. According to the rule, when multiplying powers with the same base, we add their exponents: $3^4 \times 3^x = 3^{4+x}$ .
Step 3: Write down the simplified form of the expression. The simplified expression of $3^4 \times 3^x$ is: $3^{4+x}$

Therefore, the solution to the expression $3^4 \times 3^x$ simplifies to $3^{4+x}$ .

Hence, the correct choice is $3^{4+x}$ , matching answer choice 1.

Final Answer

$3^{4+x}$

Key Points to Remember

Essential concepts to master this topic

Rule: When multiplying same bases, add the exponents together
Technique: $3^4 \times 3^x = 3^{4+x}$ by combining powers
Check: Verify base stays same and exponents are added, not multiplied ✓

Common Mistakes

Avoid these frequent errors

Multiplying the exponents instead of adding them
Don't multiply 4 × x to get $3^{4x}$ = wrong answer! This confuses the multiplication rule with the power rule. Always add exponents when multiplying same bases: $3^4 \times 3^x = 3^{4+x}$ .

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?

When you multiply powers with the same base, you're essentially counting how many times the base is multiplied by itself. $3^4 \times 3^x$ means 4 + x total multiplications of 3, so the exponent becomes 4 + x.

What if the bases were different numbers?

If the bases are different (like $2^4 \times 3^x$ ), you cannot combine them using this rule. The multiplication rule for exponents only works when the bases are identical.

Does this rule work with negative exponents too?

Yes! The rule $a^m \times a^n = a^{m+n}$ works for all exponents - positive, negative, fractions, or variables. Just add them normally.

How is this different from raising a power to a power?

Great question! $3^4 \times 3^x = 3^{4+x}$ (add exponents) is different from $(3^4)^x = 3^{4 \cdot x}$ (multiply exponents). Watch for multiplication vs. parentheses!

Can I use this rule with more than two terms?

Absolutely! For example: $3^2 \times 3^4 \times 3^x = 3^{2+4+x} = 3^{6+x}$ . Just keep adding all the exponents when the bases are the same.

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