Problem 5 - Elementary Algebra

Question

Make use of either or both the power rule for products and the power rule for powers to simplify each expression. $$ [4 t(s-5)]^{3} $$

Step-by-Step Solution

Answer

Question: Simplify the expression \([4t(s-5)]^3\) Answer: The simplified expression is \(64t^3 (s - 5)^3\).

Step 1: Apply the power rule for products

First, we'll apply the power rule for products to the given expression. The power rule for products states that \((ab)^n = a^n b^n\). In this case, \(a = 4t\) and \(b = (s - 5)\). Applying the rule, we get: \([4t(s - 5)]^3 = (4t)^3 (s - 5)^3\)

Step 2: Simplify the constants

Now, we can simplify the constant, 4, raised to the power of 3. \((4t)^3 = 4^3 t^3 = 64t^3\)

Step 3: Apply the power rule for powers

Next, let's apply the power rule for powers. The power rule for powers states that \((a^m)^n = a^{mn}\). In this case, \(a = s - 5\), \(m = 1\), and \(n = 3\). Applying the rule, we get: \((s - 5)^3 = (s - 5)^{1\cdot3} = (s - 5)^3\)

Step 4: Combine the results

Finally, we can combine the results from Steps 2 and 3 to obtain the simplified expression. \(64t^3 (s - 5)^3\) So, the simplified expression is: $$ [4t(s-5)]^3 = 64t^3 (s - 5)^3 $$

Key Concepts

Exponentiation

Exponentiation is a powerful mathematical operation that helps us express repeated multiplication in an easier way. It involves raising a base number to the power of an exponent. The expression can be written as \(a^n\), where \(a\) is the base and \(n\) is the exponent. This tells us to multiply the base, \(a\), by itself \(n\) times.

For example, in the problem given, \((4t)^3\) means we multiply \(4t\) by itself two more times, resulting in \(4t \times 4t \times 4t\). Similarly, \((s-5)^3\) means multiplying \((s-5)\) by itself twice more.

• **Base (a):** The main number you start with
• **Exponent (n):** How many times to use the base in multiplication
• **Result:** The product of the base, repeated by the number of times indicated by the exponent

Understanding exponentiation makes solving expressions involving powers much more manageable.

Simplification

Simplification in algebra involves reducing an expression to its simplest form, making it easier to understand and compute. The process often involves using rules like the power rule for products and powers. These rules allow us to break down complex expressions into more manageable parts.

In our problem, the expression \([4t(s-5)]^3\) is simplified using the power rule for products. This breaks the expression into simpler powers \((4t)^3\) and \((s-5)^3\). Then we further simplify constants, such as \(4\) raised to the third power becoming \(64\), making the expression easier to handle.

**Steps for Simplification:**

• Identify parts of the expression that can be simplified individually.
• Use mathematical rules, like power rules, to simplify components.
• Combine simplified terms for the final expression.

Simplifying expressions helps in efficiently solving algebraic problems without losing the important characteristics of the original expression.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can include numbers, variables, and operations. They are used to represent relationships and solve calculations in a general form. An algebraic expression like \(4t(s-5)\) involves a coefficient \(4\), a variable \(t\), and a binomial \((s-5)\).

• **Numbers:** Represent known values within an expression.
• **Variables:** Symbols (like \(t\) or \(s\)) that stand for unknown values.
• **Operations:** Mathematical actions like addition, subtraction, multiplication, and division that link numbers and variables.
• **Coefficients:** Numbers that multiply a variable (\(4\) in this case).

Understanding such expressions allows you to solve equations by performing operations according to algebraic rules. In the problem, recognizing that \([4t(s-5)]^3\) is an algebraic expression helps you know which rules, like the power rules, to apply to simplify it effectively."