Problem 57 - Algebra A Combined Function

Question

Use the quotient rule and simplify each expression. $$ \frac{(-4)^{6}}{(-4)^{3}} $$

Step-by-Step Solution

Answer

The simplified expression is \( -64 \).

Step 1: Understand the Problem

We need to simplify the expression \( \frac{(-4)^{6}}{(-4)^{3}} \) using the quotient rule for exponents. The quotient rule states that \( \frac{a^m}{a^n} = a^{m-n} \), where \( a \) is non-zero.

Step 2: Apply the Quotient Rule

Identify the base \( a = -4 \), and the exponents \( m = 6 \) and \( n = 3 \). By the quotient rule, we can simplify the expression as \( (-4)^{6-3} \).

Step 3: Simplify the Exponents

Calculate \( 6-3 \) which equals 3. Thus, the expression becomes \( (-4)^3 \).

Step 4: Calculate the Final Value

Now, calculate \( (-4)^3 \). This is \( -4 \times -4 \times -4 = -64 \).

Step 5: Conclude the Simplification

The simplified form of \( \frac{(-4)^6}{(-4)^3} \) is \( -64 \).

Key Concepts

Exponents

In mathematics, exponents are a shorthand way to express repeated multiplication of the same number. The base is the number that is being multiplied, while the exponent indicates how many times the base is used in the multiplication. For example, in the expression \( (-4)^6 \), \((-4)\) is the base and \(6\) is the exponent. This expression means that \((-4)\) is multiplied by itself six times: \((-4) \times (-4) \times (-4) \times (-4) \times (-4) \times (-4)\).

Using exponents simplifies calculations and makes it easier to write down long multiplications. When working with exponents, remember these key properties:

• Quotient Rule: \( \frac{a^m}{a^n} = a^{m-n} \). This rule applies when dividing two powers with the same base.
• Product Rule: \( a^m \times a^n = a^{m+n} \). This rule is for multiplying two powers with the same base.
• Power of a Power Rule: \( (a^m)^n = a^{m\cdot n} \). This rule applies when raising a power to another power.

Understanding these properties is essential for simplifying algebraic expressions efficiently.

Simplifying Expressions

Simplifying expressions is a fundamental skill in algebra that allows us to rewrite equations or terms in a more concise and manageable form. It involves combining like terms and using arithmetic operations and properties of numbers, such as the distributive property, to reduce an expression.

When simplifying expressions involving exponents, the rules of exponents are invaluable. Let's revisit the example \( \frac{(-4)^6}{(-4)^3} \):

• Identify the base and exponents: Here, \((-4)\) is the base, and the exponents \(6\) and \(3\) are associated with the numerator and denominator.
• Apply the quotient rule: This rule allows us to subtract the exponents when dividing expressions with the same base. In this case, \(6 - 3 = 3\), leading us to \((-4)^3\).
• Calculate the new expression: Simplification takes us from the fraction to a single expression by reducing the exponents, making further calculations much more straightforward.

Each step in simplifying plays a crucial role in solving algebraic expressions correctly.

Algebraic Expressions

Algebraic expressions form the foundation of algebra and consist of numbers, variables, and operations such as addition, subtraction, multiplication, and division. These expressions represent situations or values that may vary, making them highly versatile in mathematical modeling and problem-solving.

An important aspect of working with algebraic expressions is understanding how to manipulate them while maintaining their equality. This involves processes such as simplifying, factoring, and expanding expressions. Let's consider this expression: \( \frac{(-4)^6}{(-4)^3} \).

• The expression is a quotient of powers with the same base \((-4)\). By applying the quotient rule, we simplify the expression to \((-4)^{6-3}\), which is \((-4)^3\).
• The simplified expression, \((-4)^3\), represents a single term power form, which can be further calculated to yield a numerical result, namely \(-64\).
• Simplified algebraic expressions are beneficial because they are easier to interpret, analyze, and calculate, creating a clearer understanding of the mathematical relationships involved.

Grasping the manipulation of algebraic expressions enhances problem-solving skills and the ability to express mathematical ideas accurately and succinctly.