Problem 6 - Algebra 1: Concepts and Skills

Question

Write the number in decimal form. $$ 9.4 \times 10^{-5} $$

Step-by-Step Solution

Answer

The decimal form of the number \(9.4 \times 10^{-5}\) is 0.000094.

Step 1: Scientific Notation to Decimal Form

The number \(9.4 \times 10^{-5}\) is in scientific notation. In order to convert it into decimal form, the exponent of 10, which is -5 in this case, tells us how many places to move the decimal point in 9.4. A negative exponent indicates that the decimal point should be moved to the left. Therefore, move the decimal point in 9.4 five places to the left, filling in any empty places with zeroes.

Step 2: Identify the algebraic structure

Determine the type of algebraic problem.

Step 3: Apply algebraic techniques

Use factoring, expanding, or systematic methods.

Step 4: Simplify and solve

Simplify expressions and solve for unknowns.

Step 5: State the result

Write the final answer.

Step 6: Conclude with the answer

The decimal form of the number \(9.4 \times 10^{-5}\) is 0.000094.

Key Concepts

Scientific Notation

Understanding scientific notation is a crucial skill in mathematics and science, as it allows us to express very large or very small numbers in a compact form. Scientific notation is a shorthand way of writing numbers that are either very large or very small. It consists of two parts: a coefficient and a power of ten. The coefficient is a number greater than or equal to 1 and less than 10, and the power of ten is written as an exponent. For example, in the expression \(9.4 \times 10^{-5}\), \(9.4\) is the coefficient and \(-5\) is the exponent indicating the power of ten.
When converting from scientific notation to decimal form, the exponent tells us the number of places to move the decimal point.

Decimal Form

Decimal form refers to the standard way of writing numbers that include a decimal point to represent fractions. To convert a number from scientific notation to decimal form, we shift the decimal point in the coefficient a number of places equal to the absolute value of the exponent. If the exponent is negative, we move the decimal point to the left; if it's positive, to the right. In our example, we have a negative exponent \(-5\), so we move the decimal point in \(9.4\) five places to the left, which gives us \(0.000094\).
It's important to fill in any gaps with zeros to maintain the value of the original number, ensuring that the conversion between forms doesn't change the number's actual value.

Negative Exponent

When we come across a negative exponent, it might initially seem confusing. The negative sign in an exponent, such as the \(-5\) in \(9.4 \times 10^{-5}\), instructs us to take the reciprocal of the base number raised to the absolute value of the exponent. This translates to moving the decimal point to the left for negative exponents when dealing with powers of ten.
If we were to rewrite the expression without a negative exponent, it would look like this: \(\frac{1}{10^5}\) times \(9.4\), or \(9.4\) divided by \(100,000\). This interpretation reaffirms why we move the decimal to the left - because we're essentially dividing by a large number.

Algebraic Expression

An algebraic expression is a mathematical phrase that includes numbers, variables, and operation symbols. Expressions like \(9.4 \times 10^{-5}\) in scientific notation are algebraic expressions because they involve multiplication (an operation) between a numerical coefficient \(9.4\) and a power of ten (where ten is the base and \(-5\) is the exponent). Algebraic expressions are foundational in understanding mathematical relationships and solving equations, and recognizing different forms—such as scientific notation—is part of that broader understanding.