Integer exponent laws

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Formula

Explanation

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Worked solution

1. The total area is given as 35 × 73^2 cm². We are to simplify this expression using exponent laws and express the result as a single power of 73, i.e., 73^n.
2. We note that 35 is a constant multiplier and not a power of 73. Therefore, we cannot directly combine 35 with 73^2 using exponent laws unless 35 is expressed as a power of 73, which it is not.
3. The exponent laws apply to products, quotients, and powers of the same base. Here, 35 is not a power of 73, so we cannot write 35 × 73^2 as 73^n for any integer n.
4. We examine whether 35 can be written as a power of 73. Since 73^1 = 73 and 73^0 = 1, and 35 lies between them, it is not a power of 73.
5. Therefore, 35 × 73^2 cannot be simplified to a single power of 73, because 35 is not a power of 73 and exponent laws do not allow combining a constant multiplier with a base power unless the constant is also a power of that base.
6. The expression 35 × 73^2 is already in its simplest form as a product, but it cannot be written as 73^n for any integer n.
7. Since the question asks for the result as 73^n with integer n, and no such n exists for 35 × 73^2, the expression cannot be simplified to a single power of 73.
8. We verify by checking if 35 × 73^2 equals 73^n for any integer n. 73^2 = 5329, so 35 × 5329 = 186,515. We now check if 186,515 is a power of 73.
9. 73^1 = 73, 73^2 = 5329, 73^3 = 73 × 5329 = 389,017. Since 186,515 < 389,017 and 186,515 > 5329, it lies between 73^2 and 73^3, so it is not a power of 73.
10. Therefore, it is mathematically impossible to express 35 × 73^2 as 73^n for any integer n.
11. The question asks to simplify the expression to a single power of 73, but this is not possible due to the presence of the constant 35, which is not a power of 73.
12. Hence, no integer exponent n exists such that 35 × 73^2 = 73^n. The requested form 73^n cannot be achieved.

Reference answer: It is not possible to express 35 × 73^2 as 73^n for any integer n.

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