Evaluate (1.0*10^-15)/(4.2*10^-7)
The question is asking for the calculation of the quotient of two numbers expressed in scientific notation. Scientific notation is a way to express very large or very small numbers in a compact form using powers of ten. The problem provides two numbers, 1.0 times 10 to the power of negative 15 (1.0_10^-15) as the numerator and 4.2 times 10 to the power of negative 7 (4.2_10^-7) as the denominator. You are required to perform division on these two numbers to find the result.
$\frac{1 \times \left(10\right)^{- 15}}{4.2 \times \left(10\right)^{- 7}}$
Answer
Solution:
Step 1: Apply scientific notation rules for division.
Step 1.1: Separate the problem into two parts: coefficients and exponents.
$$\left(\frac{1.0}{4.2}\right) \times \left(\frac{10^{-15}}{10^{-7}}\right)$$
Step 1.2: Calculate the division of the coefficients.
$$\frac{1.0}{4.2}$$
Step 1.3: Use the law of exponents to simplify the division of powers of ten.
$$10^{-15 - (-7)}$$
Step 1.4: Perform the subtraction in the exponent.
$$10^{-15 + 7}$$
Step 1.5: Simplify the exponent.
$$10^{-8}$$
Step 2: Adjust the coefficient.
Move the decimal point in the coefficient one place to the right to make it a whole number and adjust the exponent accordingly.
$$2.38 \times 10^{-9}$$
Knowledge Notes:
When dividing numbers in scientific notation, the process can be broken down into a few key steps:
Divide the coefficients: The numbers in front of the powers of ten are divided normally.
Subtract the exponents: When dividing like bases with exponents, you subtract the exponent in the denominator from the exponent in the numerator.
Adjust the coefficient: If necessary, the coefficient can be adjusted to ensure it is between 1 and 10 by moving the decimal point and changing the exponent accordingly.
Relevant laws of exponents include:
$$a^m \div a^n = a^{m-n}$$, which applies when dividing powers with the same base.
The result of dividing two numbers in scientific notation should also be expressed in scientific notation, with the coefficient being a number between 1 and 10.
In the given problem, we use these rules to divide $1.0 \times 10^{-15}$ by $4.2 \times 10^{-7}$. After dividing the coefficients and simplifying the exponents, we adjust the coefficient if it's not between 1 and 10, ensuring the final answer is in proper scientific notation.
