Problem 28 - Algebra 1: Concepts and Skills

Question

Simplify the quotient. $$ \frac{6^{3} \cdot 6^{2}}{6^{5}} $$

Step-by-Step Solution

Answer

The simplified form of the quotient \( \frac{6^{3} \cdot 6^{2}}{6^{5}} \) is 1

Step 1: Recognize the Law of Exponents

The first step is to recognize the law of exponents. The problem is \( \frac{6^{3} \cdot 6^{2}}{6^{5}} \). When multiplying with the same base, such as \(6^{3} \cdot 6^{2}\), we add the exponents. So it will simplify to \(6^{3+2}\). This follows the law of exponents which states that \(a^{n} \cdot a^{m} = a^{n+m}\)

Step 2: Simplify the Numerator

Now simplify the numerator after applying the law of exponents. \(6^{3+2}\) equals to \(6^{5}\)

Step 3: Simplify the Quotient

Now, you are to simplify the entire quotient. Therefore, \( \frac{6^{5}}{6^{5}} \) simplifies to 1. This follows the law of exponents which states that \(a^{n}/a^{n} = 1\) for \(n \neq 0\)

Key Concepts

Simplifying Exponents

When it comes to dealing with exponents, simplification is all about making the expression as straightforward as possible. Think of exponents like shorthand: they concisely express the operation of multiplying a number by itself multiple times. Simplifying a complex expression with exponents usually involves a few steps based on exponent rules. For example, consider an expression like \( \frac{6^{3} \cdot 6^{2}}{6^{5}} \).
To simplify this, knowing that you can combine exponents with the same base when they are multiplied together is essential. Combine them by adding the exponents, turning \(6^{3} \cdot 6^{2}\) into \(6^{3+2}\), which simplifies further to \(6^{5}\). The key to simplification lies in recognizing patterns that match the exponent rules and applying them correctly to reach a more basic form of the original arithmetic expression. Use simple math at each step to avoid errors and ensure accuracy.

Exponent Rules

Exponent rules are the backbone of simplifying expressions with exponents. They are the guiding principles that structure how we handle numbers raised to powers. Here are the fundamental rules that must be mastered:

• Product of powers rule: When multiplying two exponents with the same base, you add the exponents (\( a^{n} \cdot a^{m} = a^{n+m} \) ).
• Quotient of powers rule: When dividing two exponents with the same base, you subtract the exponents in the numerator from the exponent in the denominator (\( \frac{a^{n}}{a^{m}} = a^{n-m} \) when \(meq n\); If \(m=n\), the result is 1).
• Power of a power rule: When raising an exponent to another power, you multiply the exponents (\( (a^{n})^{m} = a^{n \cdot m} \) ).
• Zero exponent rule: Any base (except 0) raised to the power of zero equals 1 (\(a^{0} = 1\))
• Negative exponent rule: A number raised to a negative exponent equals its reciprocal raised to the corresponding positive exponent (\(a^{-n} = \frac{1}{a^{n}}\)).

By applying these rules accurately, you can greatly simplify expressions with exponents. Remember that these rules help manage different scenarios of multiplication and division involving exponents, enabling a reliable method to solve problems.

Numerical Expressions

Numerical expressions are mathematical phrases that can include numbers, operators, and sometimes variables. In the case of expressions involving exponents, such as \( \frac{6^{3} \cdot 6^{2}}{6^{5}} \), the focus is on the numbers and operations, with the exponents acting as a part of them.
Solving these expressions doesn't involve guesswork but rather a set of consistent rules—exponent rules—that need to be followed. Understanding how to work with these expressions means recognizing the type of operation required (addition, subtraction, multiplication, division) and the order in which these operations should be performed. Moreover, when simplifying numerical expressions with exponents, always adhere to the proper order of operations, commonly remembered by the acronym PEMDAS ('Please Excuse My Dear Aunt Sally'), which stands for Parentheses, Exponents, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right).
In conclusion, numerical expressions with exponents become more manageable when you decode and apply the relevant exponent rules, recognize the structure of operations, and execute them in the correct sequence.