Solve 4^x × 4^x: Product Rule of Exponential Expressions

Exponential Multiplication with Same Base

4x×4x= 4^x\times4^x=

❤️ Continue Your Math Journey!

We have hundreds of course questions with personalized recommendations + Account 100% premium

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 will apply this formula to our exercise
00:14 We'll maintain the base and add the exponents together
00:18 This is the solution

Step-by-step written solution

Follow each step carefully to understand the complete solution
1

Understand the problem

4x×4x= 4^x\times4^x=

2

Step-by-step solution

To solve this problem, we will follow these steps:

  • Step 1: Identify the given expression and the operation.

  • Step 2: Apply the rule for multiplying powers with the same base.

  • Step 3: Simplify the expression to reach the final answer.

Let's proceed with the solution:
Step 1: We are given the expression 4x×4x4^x \times 4^x. Both terms have the same base of 4 and an exponent of xx.

Step 2: According to the exponent multiplication rule, am×an=am+na^m \times a^n = a^{m+n}. Here, since both bases are 4, we can simplify using this rule.

Step 3: We add the exponents, giving us:

4x×4x=4x+x=42x 4^x \times 4^x = 4^{x+x} = 4^{2x}

Therefore, the correct solution to the problem is 4x+x 4^{x+x} .

3

Final Answer

4x+x 4^{x+x}

Key Points to Remember

Essential concepts to master this topic
  • Product Rule: When multiplying same bases, add the exponents together
  • Technique: 4x×4x=4x+x=42x 4^x \times 4^x = 4^{x+x} = 4^{2x}
  • Check: Test with numbers: 42×42=16×16=256=44 4^2 \times 4^2 = 16 \times 16 = 256 = 4^4 ✓

Common Mistakes

Avoid these frequent errors
  • Multiplying the exponents instead of adding them
    Don't write 4x×4x=4x×x 4^x \times 4^x = 4^{x \times x} = wrong answer! This confuses the product rule with the power rule. Always add exponents when multiplying same bases: am×an=am+n 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 we add the exponents instead of multiplying them?

+

The product rule says when you multiply powers with the same base, you add exponents. Think of it as: 4x×4x 4^x \times 4^x means 'x factors of 4' times 'x factors of 4' = 'x+x factors of 4' total!

What's the difference between the product rule and power rule?

+

Product rule: am×an=am+n a^m \times a^n = a^{m+n} (add exponents)
Power rule: (am)n=am×n (a^m)^n = a^{m \times n} (multiply exponents)
Don't mix them up!

Can I simplify 4x+x 4^{x+x} further?

+

Yes! 4x+x=42x 4^{x+x} = 4^{2x} because x + x = 2x. Both forms are correct, but 42x 4^{2x} is the simplified version.

Does this work with different bases like 3x×4x 3^x \times 4^x ?

+

No! The product rule only works when the bases are the same. For 3x×4x 3^x \times 4^x , you'd need to use (3×4)x=12x (3 \times 4)^x = 12^x instead.

How can I remember this rule?

+

Think of exponents as counting repeated multiplication. 42×43 4^2 \times 4^3 means (4×4) × (4×4×4) = five 4's total = 45 4^5 . So 2+3=5!

🌟 Unlock Your Math Potential

Get unlimited access to all 18 Exponents Rules questions, detailed video solutions, and personalized progress tracking.

📹

Unlimited Video Solutions

Step-by-step explanations for every problem

📊

Progress Analytics

Track your mastery across all topics

🚫

Ad-Free Learning

Focus on math without distractions

No credit card required • Cancel anytime

More Questions

Click on any question to see the complete solution with step-by-step explanations

Multiplication of Powers

Reduce the Expression: 8^(3x) × 8^(3y) × 8^(2y+x)
Click to solve
Simplify the Expression: 6^a × 6^4a Using Laws of Exponents
Click to solve