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Question

Difficulty: EasyOperations with Variable Exponents

Simplify the following algebraic expression completely:

(3a5)2(3a^5)^2

What is the simplified form of this expression?

  1. 9a109a^{10}Answer
  2. B
    9a79a^7
  3. C
    6a106a^{10}
  4. D
    3a103a^{10}

Answer

The simplified expression is 9a109a^{10}.
The correct result is found by distributing the exponent of 2 to both factors inside the parentheses. The coefficient 3 is squared to become 9, and the exponent 5 is multiplied by 2 to become 10, resulting in 9a109a^{10}.

Step-by-Step Solution

1
Apply the power of a product rule by distributing the outer exponent to both the numerical coefficient and the variable inside the parentheses.
32⋅(a5)23^2 \cdot (a^5)^2
When a product is raised to a power, each independent factor inside the parentheses must be raised to that power.
2
Simplify the numerical coefficient by calculating its square.
9⋅(a5)29 \cdot (a^5)^2
The exponent 2 indicates multiplying the base by itself: 3×3=93 \times 3 = 9.
3
Apply the power of a power rule to the variable by multiplying the inner exponent by the outer exponent.
9a109a^{10}
According to exponent laws, (xm)n=xm⋅n(x^m)^n = x^{m \cdot n}, so the new exponent is 5×2=105 \times 2 = 10.

Key Concept

Power of a Product and Power of a Power Rules
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Topics

Operations with Variable Exponents

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