Question

When the number \(1.521 \times 10^{3}\) is written in ordinary decimal notation, it is expressed as _________.

Short answer

1521

Step-by-step solution

Identify the Base and Exponent

In scientific notation, a number is written in the form \(a \times 10^{b}\), where \(a\) is the base (a decimal number) and \(b\) is the exponent (an integer). In this case, the base is 1.521 and the exponent is 3.

Multiply the Base by Ten Raised to the Exponent

Now, we will multiply the base (1.521) by ten raised to the power of the exponent (3). This can be expressed mathematically as: \(1.521 \times 10^{3} = 1.521 \times (10 \times 10 \times 10)\)

Perform the Multiplication

Once we have expanded the expression, we can proceed to perform the multiplication. For this, we will simply move the decimal point in the base three places to the right as we are multiplying by a thousand (10^3 = 1000): 1.521 → 15.21 → 152.1 → 1521

Write the Result in Ordinary Decimal Notation

After the multiplication, we obtain the result in decimal notation as 1521: \(\therefore 1.521 \times 10^{3}\) in ordinary decimal notation is expressed as 1521.

Key concepts

Exponent

In mathematics, an exponent is a number that indicates how many times another number, the base, is to be multiplied by itself. This process is known as exponentiation. For example, in the expression \(2^5\), the number 2 is the base and 5 is the exponent, which means you multiply 2 by itself 5 times: \(2 \times 2 \times 2 \times 2 \times 2 = 32\).

In scientific notation, the exponent is usually a power of 10 that helps express very large or very small numbers succinctly. When calculating with scientific notation, understanding how to handle the exponent is crucial, because it tells you the number of decimal places you need to move the decimal point in the base number.

Here's a simple example: \(3 \times 10^2\) means that you would move the decimal point in the number 3 two places to the right, giving you 300. Exponents are essential in scientific and engineering calculations, as they allow for a clearer and more efficient representation of numbers.

Base (Scientific Notation)

The base in scientific notation refers to the decimal number that is used alongside the exponent of 10. In the expression \(4.56 \times 10^3\), the base is 4.56. It's important that the base is a number between 1 and 10; this is what distinguishes scientific notation from other forms of exponential representation.

In the given exercise, the number 1.521 serves as the base. When converting scientific notation to decimal notation, the base remains unchanged; you only adjust its position based on the exponent. By keeping the base within the correct range, the scientific notation provides a standardized way to represent numbers, making them easier to compare, perform calculations on, and understand at a glance.

Decimal Notation

Decimal notation is the standard form for writing numbers that we use every day. It includes a decimal point to represent fractions in a base 10 system. When dealing with scientific notation, converting to decimal notation often involves adjusting the placement of the decimal point.

The number of places the decimal point moves correlates directly with the value of the exponent. If the exponent is positive, as in our example \(1.521 \times 10^3\), the decimal moves to the right, indicating that we are working with a number larger than the base. Conversely, a negative exponent would mean moving the decimal point to the left, which is common when expressing very small numbers. It’s a straightforward process: each unit of the exponent equates to one decimal place moved.

Mathematical Multiplication

Multiplication is one of the basic operations in mathematics, involving the combination of two numbers to produce a third number called the product. In the context of scientific notation, multiplication often simplifies to shifting the decimal point a specific number of places.

When we multiply a number in scientific notation, such as \(1.521 \times 10^3\), we are actually performing a specialized form of mathematical multiplication. The multiplication by a power of 10 involves increasing the magnitude of the base number, which is done by moving the decimal point rather than performing the full calculation of the base times ten raised to the exponent. This shortcut is extremely useful in handling large-scale computations that are common in science and engineering.