Problem 24 - Algebra 1

Question

Evaluate the expression. Write fractions in simplest form. $$ \frac{8^{3} \cdot 8^{2}}{8^{5}} $$

Step-by-Step Solution

Answer

The simplified fraction is 1.

Step 1: Simplify Numerator

The numerator of the fraction is \(8^{3} \cdot 8^{2}\). In this case, apply the rule of exponents that states that when multiplying two exponential numbers with the same base, add the exponents together: \(8^{3+2}\). Which simplifies to \(8^{5}\).

Step 2: Simplify Denominator and Fraction

The denominator of the fraction is \(8^{5}\). Therefore, the whole fraction simplifies to \(\frac{8^{5}}{8^{5}}\). Apply the rule of exponents that states that when dividing two exponential numbers with the same base, subtract the exponents. Therefore, the expression becomes \(8^{5-5}\) which simplifies further to \(8^{0}\).

Step 3: Final Calculation

Finally, any nonzero number raised to the power of 0 equals 1. Therefore, \(8^{0}\) equals 1.

Key Concepts

Simplifying fractions

Simplifying fractions is an essential skill in mathematics. It involves reducing fractions to their simplest form. This means expressing the fraction with the smallest possible numerator and denominator. In our exercise, the main aim is to simplify the fraction: \[ \frac{8^{3} \cdot 8^{2}}{8^{5}} \]Understanding simplification helps mainly because dealing with smaller numbers makes calculations easier. To simplify a fraction, follow these steps:

• Find the greatest common divisor (GCD) of both the numerator and the denominator. In cases involving exponents, this step is simplified using the rules of exponents.
• Divide both the numerator and the denominator by their GCD.

For the provided exercise, applying these steps results in a basic fraction of \(\frac{8^{5}}{8^{5}}\), which simplifies further to \(8^{0}\), and ultimately to 1. Always remember, fractions equal to 1 occur when the numerator and the denominator are the same.

Exponent rules

Exponent rules are guidelines for simplifying expressions involving powers. These rules are powerful tools that allow you to work efficiently with numbers raised to powers. Generally, you need to recognize and apply the appropriate rule based on the given operation and expression. Here are three key exponent rules that were applied in the solution:

• Product of Powers Rule: This states that when multiplying two exponents with the same base, you add the exponents: \(a^{m} \cdot a^{n} = a^{m+n}\).
• Quotient of Powers Rule: When dividing two exponents with the same base, you subtract the exponents: \(\frac{a^{m}}{a^{n}} = a^{m-n}\).
• Zero Exponent Rule: Any non-zero number raised to the power of 0 is equal to 1: \(a^{0} = 1\).

These rules simplify complex expressions like the one in the exercise. They reduce them to much simpler forms, making calculations easier to handle.

Numerical expressions

Numerical expressions consist of numbers and operators, such as addition, subtraction, multiplication, and division. They also include elements like exponents and require evaluation by following the order of operations. In the given exercise, the expression to evaluate is:\[ \frac{8^{3} \cdot 8^{2}}{8^{5}} \]Evaluation means finding the simplest value or form of this expression. Here’s how you approach this:

• Identify operations: Look for how numbers are combined. In our example, multiplication and division are present.
• Apply rules: Use mathematical rules, like exponent rules, to simplify current steps. As solved earlier, we used exponent rules to deal with same base powers.
• Find the result: Once each operation is completed, simplify further to reach the final answer. Each step builds until the expression is as simple as possible.

Understanding and evaluating numerical expressions is about recognizing operations and applying the correct rules. It’s a systematic process to break down even the most complex expressions into understandable answers, such as the final result of 1 in our case.