Evaluate 1000000(1+0.07)^1
The given question is asking you to calculate the result of a mathematical expression that involves exponential growth or compound interest. Specifically, it's asking for the value of a principal amount (in this case, 1,000,000 units of currency) after being multiplied by the growth factor (1 + 0.07) raised to the power of 1. This type of calculation is often used in finance to determine the future value of an investment after a certain period of time at a given interest rate. Here 0.07 represents a growth rate or interest rate of 7%, and the exponent 1 indicates that this growth is being considered over a single time period (often one year in financial contexts).
$1000000 \left(\left(\right. 1 + 0.07 \left.\right)\right)^{1}$
Answer
Solution:
Step 1:
Combine $1$ with $0.07$ to get $1.07$. The expression becomes $1000000 \times (1.07)^1$.
Step 2:
Compute the power of $1.07$ raised to $1$, which is simply $1.07$. The expression now is $1000000 \times 1.07$.
Step 3:
Perform the multiplication of $1000000$ by $1.07$ to obtain the final result. The calculation yields $1070000$.
Knowledge Notes:
The problem at hand involves evaluating an expression with a numerical base and an exponent. The steps taken to solve this problem are part of basic arithmetic operations and the properties of exponents.
Addition: The first step involves adding two numbers, $1$ and $0.07$. This is a straightforward operation where the numbers are combined to form a new number, $1.07$.
Exponents: The next step is to deal with the exponentiation. In this case, the exponent is $1$, which means that any number raised to the power of $1$ is the number itself. Therefore, $(1.07)^1 = 1.07$.
Multiplication: The final step is to multiply the base, $1000000$, by the result of the exponentiation, $1.07$. Multiplication is one of the four fundamental operations of arithmetic and is essentially repeated addition.
The expression $1000000(1+0.07)^1$ represents a common financial calculation where an initial amount ($1000000$) is increased by a certain percentage (in this case, $7\%$). This type of calculation is often used to determine the future value of an investment or savings after applying an interest rate for a certain period (here, the period is implied to be one time period, due to the exponent $1$).
In financial mathematics, the general formula for compound interest is given by:
\[ A = P \left(1 + \frac{r}{n}\right)^{nt} \]
where:
\( A \) is the amount of money accumulated after n years, including interest.
\( P \) is the principal amount (the initial amount of money).
\( r \) is the annual interest rate (decimal).
\( n \) is the number of times that interest is compounded per year.
\( t \) is the time the money is invested for, in years.
In this problem, since the interest is compounded once and the time period is one year, the formula simplifies to the expression we evaluated.
