Algebra

3^5 × 3^2 = 3^[]

Step-by-step solution with explanation

Final Answer

The missing exponent is , giving .

Step-by-step solution

1

Identify the bases and exponents

We have two powers of 3 being multiplied together. The base is 3 for both terms, and the exponents are 5 and 2.
2

Apply the Product of Powers Rule

When you multiply two powers with the SAME base, you ADD the exponents. This is the Product of Powers rule: .
3

Add the exponents together

Simply add the two exponents: 5 plus 2 equals 7. So the missing box is filled with 7.
4

Write the final simplified expression

The completed equation is . You can verify: , , and .

Understanding this problem

Learning Insight

The Product of Powers rule works because exponents are just shorthand for repeated multiplication. means five 3s multiplied together, and means two more. Combining them gives you seven 3s multiplied together, which is .

Quick Tip

Same base multiplying? Just ADD the exponents. Same base dividing? SUBTRACT the exponents. Spotting the pattern makes these instant.

Common Mistake

Students often MULTIPLY the exponents instead of adding them, writing instead of . Multiplying exponents is the rule for a POWER of a power, like — not for multiplying two separate powers.