Problem 44 - Algebra 1

Question

EVALUATING EXPRESSIONS Evaluate the expression without using a calculator. Write the result in scientific notation and in decimal form. $$ \left(3.0 \times 10^{-3}\right)^{2} $$

Step-by-Step Solution

Answer

The solution to \((3.0 \times 10^{-3})^{2}\) is \(9.0 \times 10^{-6}\) in scientific notation and 0.000009 in decimal form.

Step 1: Evaluate the exponent

First, evaluate the expression \((3.0 \times 10^{-3})^{2}\) by applying the rule of exponents, which states that when a power is raised to another power, you multiply the exponents. The number inside the parentheses, \(3.0 \times 10^{-3}\), becomes \(3.0^{2} \times 10^{(-3*2)}\) when it's squared.

Step 2: Squaring the decimal coefficient and simplify the exponent

In the next step, square the coefficient, which is 3.0. At the same time, calculate the exponent, which is -6 after multiplying -3 by 2. Our expression now reads \(9.0 \times 10^{-6}\)

Step 3: Final representation in both scientific and decimal notation

Our final answer in scientific notation is \(9.0 \times 10^{-6}\). Now, we'll convert that to decimal form. The negative sign in the exponent means we'll move the decimal point six places to the left from its location in the number 9.0. We get 0.000009 in decimal form.

Key Concepts

Scientific Notation

Scientific notation is a method of writing very large or very small numbers in a compact form. It uses powers of 10 to express the number. The general format is \( a \times 10^{n} \) where \( a \) is a number greater than or equal to 1 and less than 10, and \( n \) is an integer. This method is particularly useful in scientific and engineering fields where such numbers are common.

For example, the scientific notation for the number 0.0000123 is \( 1.23 \times 10^{-5} \) where 1.23 is the coefficient and -5 is the exponent of 10. This notation tells us that the decimal point in the coefficient has been moved 5 places to the left to achieve the original number.

Exponents

Power of a Power Rule

The concept of exponents relates to repeated multiplication. An exponent indicates how many times a number, known as the base, is multiplied by itself. For instance, \( 5^3 \) signifies that 5 is multiplied 3 times (\( 5 \times 5 \times 5 \) ). When raising a power to another power, such as \( (3 \times 10^{-3})^2 \) in our original exercise, we multiply the exponents. This is known as the power of a power rule.

In the exercise, \( 3.0^2 \) tells us to multiply 3.0 by itself once, resulting in 9.0. The exponent \( 10^{-3} \) raised to the 2nd power, however, involves multiplying the exponents -3 and 2, yielding \( 10^{-6} \).

Decimal Form

Decimal form is the standard representation of numbers using the base-10 system. It consists of a whole number part, a decimal point, and a fractional part. Converting from scientific notation to decimal form involves shifting the decimal point to the left or right.

When the exponent in scientific notation is negative, as in \( 9.0 \times 10^{-6} \) from the exercise, it indicates we move the decimal point to the left six places, which results in 0.000009. If the exponent were positive, we'd shift the decimal to the right. Therefore, understanding the direction and quantity of these shifts is crucial for accurate conversion between scientific notation and decimal form.

Simplifying Expressions

Combining Like Terms

When simplifying mathematical expressions, the goal is to reduce them into their simplest form while maintaining their original value. This can involve combining like terms, factoring, expanding, and reducing fractions. In our example, the simplification process involves applying the rules of exponents to make the expression more manageable.

To simplify the given expression, we squared the number as separate components; the base, which is 3.0, and the exponent part, which is \(10^{-3}\). This step-by-step approach helps to prevent errors during calculations and leads to the final simple expression of \(9.0 \times 10^{-6}\), which is the simplest form of our original expression.