Simplify 365(365 root of 1.045-1)

Solution:

Step 1:

Invoke the distributive law to expand the expression: $365 \cdot \sqrt[365]{1.045} - 365 \cdot 1$

Step 2:

Compute the product of $365$ and $-1$: $365 \cdot \sqrt[365]{1.045} - 365$

Step 3:

Present the final expression in its various representations:

Exact Form: $365 \cdot \sqrt[365]{1.045} - 365$ Decimal Approximation: Approximately $0.04401953 \ldots$

Knowledge Notes:

The problem involves simplifying an expression that includes the 365th root of a number. The steps taken to simplify this expression are based on algebraic properties and arithmetic operations.

1. Distributive Property: This property states that for any real numbers $a$, $b$, and $c$, the equation $a(b + c) = ab + ac$ holds true. In the context of this problem, the distributive property is used to separate the terms within the parentheses.

2. Multiplication of Negative Numbers: When multiplying a positive number by a negative one, the result is negative. This is why $365 \times -1$ yields $-365$.

3. Roots and Radicals: The $n$th root of a number $a$, denoted as $\sqrt[n]{a}$, is a number that, when raised to the power of $n$, equals $a$. In this problem, we are dealing with the 365th root, which is a specific form of a radical expression.

4. Exact Form vs. Decimal Form: The exact form of an expression includes roots and other operations without approximating them. The decimal form, on the other hand, provides a numerical approximation of the expression, which is often necessary when dealing with irrational numbers or when a numerical answer is required.

5. Arithmetic Operations: Basic arithmetic operations, such as addition and subtraction, are used to combine or separate terms in an algebraic expression. In this problem, after applying the distributive property, the subtraction operation is used to combine the terms.