Question

How do you simplify $\left( {27,000,000} \right)\left( {0.0004} \right)$ in scientific notation?

Answer 1

Hint: We are given an expression. We have to write the equivalent number in scientific form. First, we will place the decimal after the first number to the right. Then, we will count the number of digits on the right of the decimal place. Then, write the value of count as a power of ten multiplied by the decimal number.

Complete step by step solution:
Given number is $\left( {27,000,000} \right)\left( {0.0004} \right)$

First, we will convert $27,000,000$ in scientific form.

Here, $27,000,000$ is a whole number which means the decimal will be placed at the end of the number.

Now, we will place the decimal after the first digit of the whole number by moving it 7 places to the left.

$ \Rightarrow 2.7,000,000$

Now, we will determine the exponent which is equal to the number of times we moved the decimal.

$ \Rightarrow b = 7$

Now, write the number in scientific form by substituting the values of $a$ and $b$. Since we moved the decimal to the left which means the value of exponent is positive.

$ \Rightarrow 2.7 \times {10^7}$

Now, we will convert $0.0004$ in scientific form.

Here, $0.0004$ is a decimal number and the decimal is placed at the start of the number.

Now, we will place the decimal after the first digit of the whole number by moving it 4 places to the right.

$ \Rightarrow 0004.0$

Now, we will determine the exponent which is equal to the number of times we moved the decimal.

$ \Rightarrow b = 4$

Now, write the number in scientific form by substituting the values of $a$ and $b$. Since we moved the decimal to the right which means the value of exponent is negative.

$ \Rightarrow 4.0 \times {10^{ - 4}}$

Now, multiply the numbers in scientific form.

$ \Rightarrow 2.7 \times {10^7} \times 4.0 \times {10^{ - 4}}$

Multiply the coefficients of both scientific numbers.

$ \Rightarrow 10.8 \times {10^7} \times {10^{ - 4}}$

Since, the scientific notation of a number $a \times {10^b}$ contains $1 < a < 10$. Therefore, move the decimal one place to the left of the number and write the exponent equal to the number of times we moved the decimal.

$ \Rightarrow 1.08 \times {10^1} \times {10^7} \times {10^{ - 4}}$

Now, apply the product rule of exponents, we get:

$ \Rightarrow 1.08 \times {10^{1 + 7 - 4}}$

$ \Rightarrow 1.08 \times {10^4}$

The scientific notation of $\left( {27,000,000} \right)\left( {0.0004} \right)$ is $1.08 \times {10^4}$

Note: The students must remember that the exponent of the scientific number expresses the number of placeholders present, which means the number of places the decimal will move either to the left or right to the value of base. Also, the students please note that if the power of 10 is negative, then it represents the small number in standard form whereas if the power of 10 is positive, it will represent the large number.