Multiply (322(1-(1+0.042)^-60))/0.042

Solution:

Step 1: Simplify the Numerator

• Step 1.1: Combine $1$ and $0.042$ to form $1.042$ in the expression $\frac{322(1-{\left(1.042\right)}^{-60})}{0.042}$.

• Step 1.2: Apply the negative exponent rule, which states $b^{-n}=\frac{1}{b^n}$, to rewrite the expression as $\frac{322(1-\frac{1}{{\left(1.042\right)}^{60}})}{0.042}$.

• Step 1.3: Calculate $(1.042{)}^{60}$ to find its value, which is $11.80492243$, and update the expression to $\frac{322(1-\frac{1}{11.80492243})}{0.042}$.

• Step 1.4: Compute the division of $1$ by $11.80492243$ to get $0.08471042$ and modify the expression to $\frac{322(1-1\cdot 0.08471042)}{0.042}$.

• Step 1.5: Perform the multiplication of $-1$ by $0.08471042$ to obtain $-0.08471042$.

• Step 1.6: Subtract $0.08471042$ from $1$ to get $0.91528957$ and then multiply by $322$ to simplify the numerator, resulting in $\frac{322\cdot 0.91528957}{0.042}$.

Step 2: Simplify the Expression

• Step 2.1: Multiply $322$ by $0.91528957$ to get $294.72324305$.

• Step 2.2: Divide $294.72324305$ by $0.042$ to obtain the final result, which is $7017.22007268$.

Knowledge Notes:

To solve the given problem, we need to understand several mathematical concepts:

1. Negative Exponents: The rule for negative exponents states that $b^{-n}=\frac{1}{b^n}$. This means that a number raised to a negative exponent is equal to the reciprocal of that number raised to the positive exponent.

2. Exponentiation: Raising a number to a power is a form of exponentiation. For example, $b^n$ means multiplying the base $b$ by itself $n$ times.

3. Arithmetic Operations: Basic arithmetic operations include addition, subtraction, multiplication, and division. These operations are used in sequence to simplify expressions.

4. Order of Operations: When simplifying mathematical expressions, it's important to follow the correct order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right)).

5. Simplifying Fractions: To simplify a fraction, you perform the division of the numerator by the denominator. This can involve multiplication or division of large numbers or decimals.

In this problem, we applied these concepts to simplify the given expression step by step, ensuring that we followed the order of operations and correctly applied the rules for negative exponents and arithmetic operations.