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Difficulty: MediumOperations with Variable Exponents

The length of a rectangular solar panel is given by the expression 7a4b−27a^4 b^{-2} and its width is given by 6a2b56a^2 b^5. Which of the following expressions represents the area of the solar panel?

  1. 42a6b342a^6 b^3Answer
  2. B
    42a8b−1042a^8 b^{-10}
  3. C
    42a6b−742a^6 b^{-7}
  4. D
    13a6b313a^6 b^3

Answer

The area is represented by the expression 42a6b342a^6 b^3.
The correct answer is 42a6b342a^6 b^3. To find the area of the rectangle, you multiply its length and width: (7a4b−2)(6a2b5)(7a^4 b^{-2})(6a^2 b^5). The coefficients are multiplied together (7×6=427 \times 6 = 42). For the variables, the product rule states that when multiplying terms with the same base, you add their exponents. Thus, for base aa, 4+2=64 + 2 = 6 to get a6a^6, and for base bb, −2+5=3-2 + 5 = 3 to get b3b^3. Combining these yields 42a6b342a^6 b^3.

Step-by-Step Solution

1
Identify the formula for the area of a rectangle and set up the expression.
Area = length \times width = (7a^4 b^{-2})(6a^2 b^5)
The area of a rectangle is found by taking the product of its length and width dimensions.
2
Multiply the numerical coefficients.
7×6=427 \times 6 = 42
Constants must be multiplied independently from the variable bases.
3
Apply the product rule for exponents to the variables with identical bases.
a^{4+2} = a^6 and b^{-2+5} = b^3
When multiplying identical bases, their exponents must be added.
4
Combine the calculated coefficient and variables into a single expression.
42a6b342a^6 b^3
This forms the final simplified product representing the area.

Key Concept

Operations with Variable Exponents
Estimated Time:1m 0s
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Topics

Operations with Variable Exponents

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