Multiply (322(1-(1+0.042)^-60))/0.042
This question presents a mathematical expression that requires one to perform a series of operations to arrive at a solution. Specifically, the task is to multiply a certain number, 322, by the result of subtracting the result of another operation from 1. The inner operation involves taking 1 plus 0.042 to the negative 60th power (which implies using the formula for a geometric series or compound interest). Finally, this result is further subtracted from 1, and the outcome is divided by 0.042 before being multiplied by 322. The overall expression appears to be related to financial calculations, perhaps calculating the present value or payment of an annuity at a certain interest rate.
$\frac{322 \left(\right. 1 - \left(\left(\right. 1 + 0.042 \left.\right)\right)^{- 60} \left.\right)}{0.042}$
Answer
Solution:
Step 1: Simplify the Numerator
Step 1.1: Combine $1$ and $0.042$ to form $1.042$ in the expression $\frac{322 \left( 1 - \left(1.042\right)^{-60} \right)}{0.042}$.
Step 1.2: Apply the negative exponent rule, which states $b^{-n} = \frac{1}{b^n}$, to rewrite the expression as $\frac{322 \left( 1 - \frac{1}{\left(1.042\right)^{60}} \right)}{0.042}$.
Step 1.3: Calculate $(1.042)^{60}$ to find its value, which is $11.80492243$, and update the expression to $\frac{322 \left( 1 - \frac{1}{11.80492243} \right)}{0.042}$.
Step 1.4: Compute the division of $1$ by $11.80492243$ to get $0.08471042$ and modify the expression to $\frac{322 \left( 1 - 1 \cdot 0.08471042 \right)}{0.042}$.
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:
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.
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.
Arithmetic Operations: Basic arithmetic operations include addition, subtraction, multiplication, and division. These operations are used in sequence to simplify expressions.
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)).
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.
