Question

EVALUATING EXPRESSIONS Evaluate the expression without using a calculator. Write the result in scientific notation and in decimal form. $$ \left(9 \times 10^{3}\right)^{2} $$

Short answer

The result of \((9 \times 10^{3})^{2}\) in scientific notation is \(81 \times 10^{6}\) and in decimal form it is 81000000.

Step-by-step solution

Understanding the Expression

The given expression is \((9 \times 10^{3})^{2}\). Here, the \(9 \times 10^{3}\) portion represents a number in scientific notation. The raised to power 2 (\(^{2}\)) indicates that we must square this number.

Square the Number

When we square the number, we square each component separately. Squaring the coefficient part (9), we get \(9^{2} = 81\). When squaring the \(10^{3}\) part, remember to multiply the exponents. Hence, \( (10^{3})^{2} = 10^{6}\).

Write the Final Expression in Scientific Notation

The final result, in scientific notation, is the product of the results from step 2, that is \(81 \times 10^{6}\).

Convert the Scientific Notation to Decimal Form

To convert from scientific notation to decimal form, we move the decimal point of 81 six places to the right (Because the exponent of 10 is 6). This gives us the decimal form of the result: 81000000.

Key concepts

Scientific Notation

Scientific notation is a method of writing very large or very small numbers in a compact form. It is especially useful in fields like science and engineering where such numbers are common.

For instance, the number 450,000,000 can be difficult to read, but in scientific notation, it becomes a more manageable \(4.5 \times 10^8\). Scientific notation has two parts: a coefficient and an exponent. The coefficient is a number typically between 1 and 10, and the exponent indicates how many times the number should be multiplied by 10.

When evaluating expressions in scientific notation, each component is handled separately, as seen in the original exercise where \(9 \times 10^3\) is squared. Converting to and from scientific notation requires an understanding of place value and the use of exponents, which represent the number of times 10 is used as a multiplying factor. It's important to be comfortable with moving the decimal point in accordance with the exponent when converting between scientific notation and decimal form.

Squaring Numbers

Squaring a number essentially means multiplying that number by itself. It is an example of an exponential function, where the base (the number being multiplied) is raised to the power of 2. The notation \(x^2\) represents the square of \(x\).

This concept is frequently seen in geometry, physics, and algebra among various fields. For example, the area of a square is found by squaring the length of its sides. In our exercise, squaring the coefficient 9 results in \(9^2 = 81\).

h4. Applying Squaring to Place Value
When squaring whole numbers, it is essential to consider the place value. Squaring a number like 10, which represents \(10^1\), results in \(100\), or \(10^2\), since 'squared' means 'to the power of 2'. It's vital for students to recognize how squaring affects both the coefficient and the exponent in terms of growth in value when dealing with scientific notation.

Exponential Notation

Exponential notation is a concise way to represent repeated multiplication of a number by itself. It is written with a base raised to an exponent, as in \(a^n\), which tells us to multiply the base \(a\) by itself \(n\) times.

In the context of scientific notation, the base is often 10, and the exponent can be any integer, positive or negative, reflecting the magnitude of the number. For large numbers, the exponent is positive, and for small numbers (like fractions or decimals), the exponent is negative.

h4. Exponent Rules
When calculating with exponential notation, it's critical to understand exponent rules. For instance, when an exponential number is raised to another power, such as \( (10^3)^2 \), you multiply the exponents, resulting in \( 10^{3 \times 2} = 10^6 \). This concept is fundamental when evaluating expressions with exponents and is a cornerstone for higher level math like calculus, where exponents play a key role in differentiation and integration.