Question

EVALUATING EXPRESSIONS Evaluate the expression without using a calculator. Write the result in scientific notation and in decimal form. $$ \left(3 \times 10^{-2}\right)^{4} $$

Short answer

The expression \(\left(3 \times 10^{-2}\right)^{4}\) equals \(81 \times 10^{-8}\) in scientific notation and 0.00000081 in decimal form.

Step-by-step solution

Evaluate the Base of the Expression

First, separate the scientific notation term and the integer in the base of the expression. Specifically \(\left(3 \times 10^{-2}\right)^{4}\) is equivalent to \(3^{4} \times (10^{-2})^{4}\). Now you can calculate these two terms separately.

Evaluate the Powers

Calculate \(3^{4}\), which is 3 multiplied by itself 4 times. Next, calculate \((10^{-2})^{4}\), which equals \(10^{-8}\). The exponent is calculated as -2 (the original exponent) times 4 (the outside exponent). Combine both results.

Convert the Result into Decimal Form

Now, put the result from the previous step in decimal form. The scientific notation \(10^{-8}\) represents the number that is obtained by moving the decimal point in '1' 8 places to the left, which produces a very small decimal number.

Key concepts

Scientific Notation

Scientific notation is a method of expressing numbers that are too large or too small to be conveniently written in decimal form. It is commonly used by scientists, engineers, and mathematicians to handle the vast range of values they encounter in the natural world. For example, the distance between stars, the size of microscopic organisms, or the speed of light.

In scientific notation, a number is written as the product of two factors: a decimal number with a magnitude between 1 and 10, and a power of 10. The general form is \( a \times 10^{n} \), where \(|a|\) is between 1 and 10, and \(|n|\) is an integer. This format conveniently portrays the number's magnitude through the exponent \(|n|\), allowing us to easily compare and calculate with very large or very small values. For instance, \(3 \times 10^{-2}\) indicates that the decimal point in the number 3 is moved 2 places to the left, resulting in a value of 0.03.

Exponentiation

Exponentiation is a mathematical operation involving two numbers, the base and the exponent. It is denoted as \(b^{n}\), where 'b' is the base and 'n' is the exponent or power. In the context of our exercise, the base is \(3 \times 10^{-2}\) and the exponent is 4, indicating the base should be multiplied by itself four times.

Understanding exponentiation requires recognizing two main rules:

• The product of powers rule states that when multiplying two of the same bases with different exponents, you add the exponents.
• The power of a power rule instructs us that when raising a power to another power, you multiply the exponents.

Applying these laws to our example \(3^{4} \times (10^{-2})^{4}\), we can simplify the mathematical process effectively to find our product.

Decimal Form

Decimal form is a standard way of writing numbers that includes a decimal point. The decimal point helps us to precisely indicate values less than one or to represent fractions in an easy-to-understand manner. To convert scientific notation to decimal form, you move the decimal point to the right or left according to the sign of the exponent. A positive exponent moves the decimal point to the right, creating a larger number, while a negative exponent moves it to the left, creating a smaller number.

For the given exercise, the scientific notation \(10^{-8}\) tells us that the decimal in the number 1 is moved 8 places to the left, resulting in \(0.00000001\). Therefore, when converting from scientific notation to decimal form, it’s essential to recognize the role of the exponent and move the decimal point accordingly. This conversion process is critical for visualizing the true size of the number, especially when scientific notation can seem abstract.