It takes 10 years for a $100 monthly bank deposit to amount to $15,528.23 when invested at 5% interest compounded monthly. Express the amount $15,528.23 in a firm of a×10^n where a is single digit.

Key words for this: "single digit"

2×10⁴ is the correct answer.

Thank you

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To find the amount $15,528.23 in the form of a×10^n, we need to express it in scientific notation.

Scientific notation is a way of writing large or small numbers using powers of 10. The general format is a × 10^n, where "a" is a single digit and "n" represents the exponent of 10.

To express $15,528.23 in scientific notation, we need to determine the value of "a" and "n".

Let's start by reformatting the given amount to have 2 decimal places for consistency: $15,528.23.

The value of "a" can be determined by considering the digit before the decimal point, which is 1 in this case.

Now, to calculate the value of "n", we need to determine the number of times we would need to multiply 10 by itself to obtain the total amount.

In this case, it is given that the $100 monthly deposit amounts to $15,528.23 in 10 years, with interest compounded monthly at 5%.

First, calculate the total number of compounding periods over the 10-year period: 10 years × 12 months/year = 120 months.

Now, we can use the compound interest formula to find the value of "n":

A = P(1 + r/n)^(nt)

Where:
A = Final amount ($15,528.23)
P = Principal amount ($100)
r = Annual interest rate (5% or 0.05)
n = Number of times interest is compounded per year (12)
t = Number of years (10)

15,528.23 = 100(1 + 0.05/12)^(12*10)

Now, we can solve for the value of "n" by taking the logarithm (base 10) of both sides of the equation:

log(15,528.23) = log(100(1 + 0.05/12)^(12*10))

Using a calculator, the left side equals approximately 4.19.

Now, let's simplify the right side of the equation:

log(100(1 + 0.05/12)^(12*10))
log(100(1.0041667)^120)
log(100(1.6487213))
log(164.87213)
2.218

Therefore, we have:

4.19 = 2.218n

Dividing both sides of the equation by 2.218, we get:

n = 4.19 ÷ 2.218
n ≈ 1.886

So, the value of "n" is approximately 1.886.

Now, we can express the amount $15,528.23 in the form of a×10^n:

$15,528.23 ≈ 1.552823 × 10^4

Thus, the amount $15,528.23 can be expressed in the form of a×10^n as approximately 1.552823 × 10^4.

a

is single a digit.