Standard Form Calculation Explained

To find the standard form of the given expression, we need to convert each number into its standard form (also known as scientific notation) first. Then, we will perform the multiplication and division operations using the rules of exponents.

Understanding Standard Form

Standard form or scientific notation is a way of writing very large or very small numbers concisely. A number is written in standard form when it is expressed as:

\(a \times 10^b\)

Where:

• \(a\) is a number greater than or equal to 1 and less than 10 (\(1 \le a < 10\)).
• \(b\) is an integer (positive for large numbers, negative for small numbers).

Converting Numbers to Standard Form

Let's convert each number in the expression \(\dfrac{36000,000 × 0.0000025}{0.0000005}\) into its standard form:

• Converting \(36,000,000\):

  To get \(a\) between 1 and 10, we move the decimal point from the end to after the first non-zero digit. For \(36,000,000\), the decimal point moves 7 places to the left.

  \(36,000,000 = 3.6 \times 10^7\)

• Converting \(0.0000025\):

  To get \(a\) between 1 and 10, we move the decimal point from its current position to after the first non-zero digit. For \(0.0000025\), the decimal point moves 6 places to the right.

  \(0.0000025 = 2.5 \times 10^{-6}\)

• Converting \(0.0000005\):

  Similarly, for \(0.0000005\), the decimal point moves 7 places to the right.

  \(0.0000005 = 5.0 \times 10^{-7}\)

Performing the Calculation

Now, substitute these standard form values back into the original expression:

\(\dfrac{(3.6 \times 10^7) \times (2.5 \times 10^{-6})}{5.0 \times 10^{-7}}\)

Step 1: Simplify the numerator (multiplication)

Multiply the numerical parts and the powers of 10 separately:

\((3.6 \times 2.5) \times (10^7 \times 10^{-6})\)

For the numerical part:

\(3.6 \times 2.5 = 9.0\)

For the powers of 10, use the rule \(10^m \times 10^n = 10^{(m+n)}\):

\(10^7 \times 10^{-6} = 10^{(7 + (-6))} = 10^{(7 - 6)} = 10^1\)

So, the numerator simplifies to:

\(9.0 \times 10^1\)

Step 2: Perform the division

Now, divide the simplified numerator by the denominator:

\(\dfrac{9.0 \times 10^1}{5.0 \times 10^{-7}}\)

Divide the numerical parts and the powers of 10 separately:

\(\left(\dfrac{9.0}{5.0}\right) \times \left(\dfrac{10^1}{10^{-7}}\right)\)

For the numerical part:

\(\dfrac{9.0}{5.0} = 1.8\)

For the powers of 10, use the rule \(10^m / 10^n = 10^{(m-n)}\):

\(\dfrac{10^1}{10^{-7}} = 10^{(1 - (-7))} = 10^{(1 + 7)} = 10^8\)

Combining these results, the expression simplifies to:

\(1.8 \times 10^8\)

Final Answer in Standard Form

The standard form of the given expression is \(1.8 \times 10^8\).

Comparing with Options

Let's compare our calculated standard form with the given options:

Our result, \(1.8 \times 10^8\), matches Option 1.