Final answer:
The expression (3.8 x 2^-5 x 9^0)^-2 * (2^-2/3^3)^4 is solved by simplifying within the parentheses, raising a power to a power, and combining exponents properly. In scientific notation, multiplication involves adding exponents, while division subtracts exponents, facilitating the simplification process.
Step-by-step explanation:
To solve the expression (3.8 x 2^-5 x 9^0)^-2 * (2^-2/3^3)^4, we must follow the order of operations and apply the rules of exponents methodically. First, we simplify within the parentheses, noting that any number raised to the power of 0 is 1, so 9^0 becomes 1. Next, we can simplify the expression by using the properties of exponents, specifically that when we raise a power to a power we multiply the exponents, and when we divide powers with the same base we subtract the exponents. For example, in (2^-2/3^3)^4, the exponents -2 and 3 are both raised to the power of 4, so we multiply them to get 2^-8 and 3^12 respectively. The expression outside the parentheses starts with a negative exponent, which means we take the reciprocal of the base and change the sign of the exponent. Simplifying every step, we obtain the final expression in proper form.
When dealing with scientific notation, it's important to understand that in multiplication, we add the exponents of powers of 10, whereas in division, we subtract the exponents. This simplifies the process considerably and allows for calculations without converting all the numbers from scientific notation to standard form. An example of this would be evaluating expressions that involve multiplication or division of numbers in scientific notation. Furthermore, when converting a non-standard scientific notation number to standard form, the exponent n must be adjusted accordingly to maintain the same value.