Final answer:
When dealing with exponents, the power-to-power rule is applied by multiplying the exponents inside the parentheses. Negative exponents represent division or inversion. In scientific notation, arithmetic with powers of ten is simplified by managing coefficients and adding exponents.
Step-by-step explanation:
When dealing with exponents and parentheses, the rules for exponentiation apply to everything inside the parentheses. For example, when raising an expression such as (5³)⁴ to a power, we apply the power-to-power rule, multiplying the exponents together to get 5³×4 = 5¹². Similarly, when multiplying powers of 10, we simply add their exponents as shown in the expression (3 × 10⁵) × (2 × 10⁰) = 6 × 10⁵+0 = 6 × 10⁵.
When we invert an exponentiation, a negative exponent signifies division, as in x⁻¹ = 1/x. Therefore, for an equation like 3⁴ × 3⁻⁴, using the rule of adding exponents, we get 3⁴+(-4) = 3⁰, which equals 1. This demonstrates that 3⁻⁴ is equivalent to 1/3⁴.
Scientific notation makes arithmetic operations involving powers of ten straightforward. When multiplying, we manage the coefficients and then add the exponents to simplify calculations.