Problem 65 - Algebra 1: Concepts and Skills

Question

Perform the indicated operation. Write the result in scientific notation. (Lesson 8.5). $$ \frac{8 \times 10^{-3}}{5 \times 10^{-5}} $$

Step-by-Step Solution

Answer

The result is \( 1.6 \times 10^{2} \)

Step 1: Divide the coefficients

Divide the coefficient of the numerator (8) by the coefficient of the denominator (5). That is, \( \frac{8}{5} = 1.6 \)

Step 2: Subtract the exponents

Subtract the exponent of the denominator (-5) from the exponent of the numerator (-3). That is, -3 - (-5) = -3 + 5 = 2

Step 3: Put the result back into scientific notation

Combine the result of the divided coefficients and the subtracted exponents back into scientific notation. The result is \( 1.6 \times 10^{2} \)

Key Concepts

Division of Exponents

Working with exponents might seem tricky at first, but understanding how to divide them is essential in mathematics, especially when dealing with scientific notation. When dividing numbers in scientific notation, the exponents are part of the operation. The rule for dividing exponents is straightforward:

• Subtract the exponent of the divisor from the exponent of the dividend.

For example, in the exercise: \[\frac{8 \times 10^{-3}}{5 \times 10^{-5}}\] the exponents are -3 and -5. Subtract the exponent of the denominator (-5) from the exponent of the numerator (-3) to get the result: \[-3 - (-5) = -3 + 5 = 2.\] This simplification is a crucial step in division involving exponents and helps maintain the rules of powers.

Exponents

Exponents are a shorthand way to express repeated multiplication. They help make large or small numbers easily manageable, especially in scientific notation. For example, \(10^{-3}\) is an exponent that tells us to multiply 10 by itself with an inverse process, which is quite handy for representing particularly small numbers:

• \(10^{-3} = \frac{1}{1000}\).

Exponents follow certain basic rules that help with operations:

• Product of Powers: When multiplying terms with the same base, you add the exponents.
• Quotient of Powers (Division Rule): When dividing terms with the same base, subtract the exponents as shown in the example above.

These rules make exponents a powerful tool for simplifying mathematical expressions.

Mathematical Operations

Mathematical operations like addition, subtraction, multiplication, and division form the core of mathematical problem-solving, and they follow certain principles even when working with complex expressions like scientific notation. In scientific notation:

• Multiplication and Division: Handle the coefficients separately from the powers of 10. Perform standard arithmetic operations for coefficients and use the rules of exponents for the powers of 10.
• Ensure the final expression conforms to the format \(a \times 10^n\) with 1 ≤ |a| < 10.

Using these operations correctly ensures that even complex numbers or small values are represented efficiently and accurately, as seen through our example: \[ \frac{8 \times 10^{-3}}{5 \times 10^{-5}} = 1.6 \times 10^{2}.\] This precision is why scientific notation and understanding its mathematical operations are valuable in both academic and real-world scenarios.