Problem 58 - Algebra A Combined Function

Question

Use the quotient rule and simplify each expression. $$ \frac{(-6)^{13}}{(-6)^{11}} $$

Step-by-Step Solution

Answer

The expression simplifies to 36.

Step 1: Apply the Quotient Rule

The quotient rule for exponents states that \( \frac{a^m}{a^n} = a^{m-n} \). Use this rule to simplify the expression \( \frac{(-6)^{13}}{(-6)^{11}} \). This will give you \((-6)^{13-11}\).

Step 2: Simplify the Exponent

Perform the subtraction in the exponent: \(13 - 11 = 2\). Therefore, the expression becomes \((-6)^2\).

Step 3: Calculate the Final Expression

Now, calculate \((-6)^2\). Since \((-6)\times (-6) = 36\), the simplified value is \(36\).

Key Concepts

Exponents

Exponents are a way to represent repeated multiplication of the same factor. If you see an expression like \(a^n\), this simply means you are multiplying \(a\) by itself \(n\) times. For example, \((-6)^{13}\) means that you multiply \(-6\) by itself 13 times. Exponents make it efficient to write and work with large multiplication problems without needing to list out each factor.
When dealing with exponents, there are several rules that help simplify expressions. The quotient rule is particularly useful when dividing like bases, as it allows you to subtract the exponents: \(\frac{a^m}{a^n} = a^{m-n}\). This is because dividing \(a^m\) by \(a^n\) cancels \(n\) factors of \(a\) from the numerator, reducing the power of \(a\) in the expression.

Simplifying expressions

Simplifying expressions involves reducing a mathematical expression to its simplest form. This often means using various algebraic rules to combine like terms or reduce fractions.
For example, in the expression \(\frac{(-6)^{13}}{(-6)^{11}}\), you apply the quotient rule to simplify it. Here, both the numerator and the denominator have the same base, \(-6\). By subtracting the exponents \(13 - 11\), we simplify the expression to \((-6)^2\).
Simplifying makes expressions easier to understand and work with when solving equations or evaluating them numerically. Always aim to simplify expressions as much as possible to make calculations and comparisons more straightforward.

Algebraic operations

Algebraic operations involve the basic mathematical functions like addition, subtraction, multiplication, and division, as well as more complex actions such as working with exponents and roots. These operations form the backbone of algebra and are used to manipulate algebraic expressions.
When working with expressions like \((-6)^2\), you multiply \(-6\) by itself. In this case, \((-6) \times (-6) = 36\), because multiplying two negative numbers yields a positive result. Understanding how to perform these basic operations is essential for effectively managing algebraic manipulations.
Mastering algebraic operations allows you to confidently handle expressions, equations, and problems across different areas of mathematics. Practice is key to becoming proficient in recognizing which operation or rule to apply to simplify or solve a given problem.