Simplify the Expression: 3² × 3³ Using Laws of Exponents

Exponent Multiplication with Same Base

Simplify the following equation:

32×33= 3^2\times3^3=

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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)
00:13 We will apply this formula to our exercise
00:17 We will then proceed to add up the exponents and raise them to this power
00:20 This is the solution

Step-by-step written solution

Follow each step carefully to understand the complete solution
1

Understand the problem

Simplify the following equation:

32×33= 3^2\times3^3=

2

Step-by-step solution

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

  • Step 1: Identify the base and exponents
  • Step 2: Apply the exponent multiplication rule
  • Step 3: Perform the calculations

Let's work through each step:
Step 1: We have 32 3^2 and 33 3^3 . Both have the same base, which is 3.
Step 2: According to the exponent multiplication rule am×an=am+n a^m \times a^n = a^{m+n} , we add the exponents:
2+3=5 2 + 3 = 5 .
Step 3: Rewrite the expression as a single power:
32×33=32+3=35 3^2 \times 3^3 = 3^{2+3} = 3^5 .

Therefore, the simplified expression is 35\boldsymbol{3^5}, which corresponds to choice 2.

3

Final Answer

35 3^5

Key Points to Remember

Essential concepts to master this topic
  • Rule: When multiplying powers with same base, add the exponents
  • Technique: 32×33=32+3=35 3^2 \times 3^3 = 3^{2+3} = 3^5 using exponent addition
  • Check: Verify by expanding: 32=9 3^2 = 9 , 33=27 3^3 = 27 , so 9×27=243=35 9 \times 27 = 243 = 3^5 ✓

Common Mistakes

Avoid these frequent errors
  • Multiplying the exponents instead of adding them
    Don't calculate 32×3=36 3^{2 \times 3} = 3^6 = 729! This gives a completely different answer. When bases are the same, the rule is to add exponents, not multiply them. Always use am×an=am+n a^m \times a^n = a^{m+n} .

Practice Quiz

Test your knowledge with interactive questions

\( \)

Simplify the following equation:

\( 5^8\times5^3= \)

FAQ

Everything you need to know about this question

Why do we add the exponents instead of multiplying them?

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Think about what exponents mean! 32×33 3^2 \times 3^3 means (3 × 3) × (3 × 3 × 3). That's 3 multiplied by itself 5 times total, which equals 35 3^5 .

What if the bases are different, like 23×32 2^3 \times 3^2 ?

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You cannot combine exponents when bases are different! You must calculate each power separately: 23=8 2^3 = 8 and 32=9 3^2 = 9 , so the answer is 8 × 9 = 72.

How can I remember the exponent multiplication rule?

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Use the phrase "Same base, add the race!" The "race" refers to the exponents. When bases match, you add the exponents together.

Can I just multiply 3² and 3³ directly without using the rule?

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Yes, but it's much harder! 32=9 3^2 = 9 and 33=27 3^3 = 27 , so 9×27=243 9 \times 27 = 243 . Using the rule gives 35=243 3^5 = 243 much faster!

What does 35 3^5 actually equal?

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35=3×3×3×3×3=243 3^5 = 3 \times 3 \times 3 \times 3 \times 3 = 243 . But for this problem, leaving the answer as 35 3^5 is the simplified form they're looking for.

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