Problem 101 - Algebra A Combined Function

Question

Evaluate each expression using exponential rules. Write each result in standard form. $$ \frac{8 \times 10^{-1}}{16 \times 10^{5}} $$

Step-by-Step Solution

Answer

The expression evaluates to \( 5 \times 10^{-7} \) in standard form.

Step 1: Simplify the Coefficients

The expression \( \frac{8}{16} \) simplifies to \( \frac{1}{2} \) since both numbers are divisible by 8. Therefore, the coefficient part simplifies to \( \frac{1}{2} \).

Step 2: Apply Exponential Rules

Given \( 10^{-1} \) in the numerator and \( 10^{5} \) in the denominator, we use the law of exponents: \( \frac{a^m}{a^n} = a^{m-n} \). So, \( 10^{-1 - 5} = 10^{-6} \).

Step 3: Combine Simplified Parts

Combine the simplified coefficient \( \frac{1}{2} \) and the result from the exponents \( 10^{-6} \) to form \( \frac{1}{2} \times 10^{-6} \).

Step 4: Convert to Standard Form

To express \( \frac{1}{2} \times 10^{-6} \) in standard form, calculate \( \frac{1}{2} = 0.5 \). Thus, the expression in standard form is \( 0.5 \times 10^{-6} = 5 \times 10^{-7} \).

Key Concepts

Standard Form

Standard form is a way of writing numbers that makes them easier to read and compare, especially when dealing with very large or very small numbers. It is often used in scientific notation to express numbers neatly. In standard form, a number is written as the product of two parts:

• A decimal number, usually between 1 and 10.
• And a power of ten.

For example, the number 500 can be written as \(5 \times 10^2\). This form helps manage numbers by focusing on their scale and size rather than just their digits.
In the context of your exercise, converting \( \frac{1}{2} \times 10^{-6} \) to standard form means turning 0.5 into \(5\) and adjusting the exponent from \(10^{-6}\) to \(10^{-7}\) after taking the decimal into account. Thus, it becomes \(5 \times 10^{-7}\), making it much simpler to understand and use.

Law of Exponents

The law of exponents is a set of rules used to simplify expressions involving powers of the same base. One key rule is the quotient rule, which is applied when dividing two exponential terms with the same base. This law states that you subtract the exponents:

• \( \frac{a^m}{a^n} = a^{m-n} \)

In your exercise, the base is 10. You have \(10^{-1}\) in the numerator and \(10^5\) in the denominator. Applying the law, you subtract the exponent in the denominator from the one in the numerator:
\( 10^{-1 - 5} = 10^{-6} \).
These rules simplify complex exponent problems by reducing them to more manageable expressions, letting you solve them with ease.

Simplifying Fractions

Simplifying fractions is the process of reducing them to their simplest form. This involves dividing the numerator and denominator by their greatest common factor (GCF).
For the fraction \(\frac{8}{16}\), the GCF is 8, since 8 is the largest number that divides both 8 and 16 evenly. So, we divide both the numerator and the denominator by 8:

• \(\frac{8}{16} = \frac{8 \div 8}{16 \div 8} = \frac{1}{2}\)

Once simplified, the fraction becomes \(\frac{1}{2}\) which is much easier to work with in calculations. Simplifying helps in keeping the numbers manageable and the calculations less error-prone.