Final answer:
To ensure positive exponents while simplifying expressions, one must add exponents when multiplying, subtract when dividing, and multiply when raising powers. The goal is to maintain positive exponents in the final result, sometimes requiring the expression to be rewritten as the reciprocal of the base.
Step-by-step explanation:
When simplifying expressions with exponents and ensuring positive exponents, we follow certain algebraic rules for exponent operations. To multiply expressions with the same base, we add the exponents. For division, we subtract exponents. When raising a power to another power, we multiply the exponents. It is important to treat negative exponents the same as positive ones in arithmetic operations, and if the result has a negative exponent, we rewrite it as a positive exponent by taking the reciprocal of the base.
For example, to simplify 5,1 × 10³ × 4,2 × 10⁴, you multiply the digit terms normally to get 21,42 and add the exponents of the exponential terms to end up with 2,142 × 10⁸, ensuring that the final result has positive exponents.
When dividing, such as 1,000,000 ÷ 1000, you get 10⁶ ÷ 10³ which simplifies to 10³ by subtracting the exponents. Similarly, to raise an expression inside parentheses to a power, such as (5³)⁴, you multiply the exponents to get 5¹².