Problem 45 - Algebra 1: Concepts and Skills

Question

Simplify the expression. \((-r s t)^{5}\)

Step-by-Step Solution

Answer

The simplified form of the given expression is \(-r^5 s^5 t^5\).

Step 1: Understand the given expression

The given expression is \((-r s t)^{5}\). This is an algebraic expression where \(r\), \(s\), and \(t\) are variables, and the whole expression is raised to the power of 5.

Step 2: Apply exponent to each part of the expression

Applying the exponent to each part of the expression, we get \(-1^5\cdot r^5\cdot s^5\cdot t^5\).

Step 3: Simplify the first part of the expression

Raising \(-1\) to an odd power results in \(-1\). So \(-1^5\) simplifies to \(-1\).

Step 4: Present the final simplified expression

Combining all of these, we get the final simplified expression is \(-r^5 s^5 t^5\).

Key Concepts

Algebraic Expressions

Algebraic expressions are a fundamental aspect of algebra that incorporate numbers, variables, and operations. Variables are symbols, typically letters like \(r\), \(s\), and \(t\), which stand in for unknown or variable quantities within the expression. They allow us to write equations that can express relationships and changes.
Consider our example, \((-r s t)^{5}\). Here, \(r\), \(s\), and \(t\) are variables grouped together as a product, and together they form part of a larger expression. An algebraic expression can consist of numerous operations and groupings, such as sums, products, and powers. This expression is particularly focused on the exponentiation of a product of variables, a common scenario in algebra.
It's important to understand that, within an algebraic expression, each part can represent a real-world quantity, and interpreting them correctly is crucial for problem-solving. Whether you are working on simplifying or evaluating them, the skill to handle algebraic expressions is vital.

Simplification

Simplification is the process of changing a mathematical expression into its simplest form. In algebra, this often means reducing expressions while maintaining their original value. For example, simplifying \((-r s t)^{5}\) involves using rules of exponents and arithmetic to condense and present the expression in its most efficient form.
When simplifying:

• Apply any coefficients directly.
• Use exponent rules to distribute powers.
• Combine like terms and apply any arithmetic simplifications, such as signs.

You begin with applying the exponent to each factor within the parenthesis: the number, and each variable (\(-r\), \(s\), \(t\)).
After assigning the power, the expression becomes \(-1^5 \, r^5 \, s^5 \, t^5\) due to exponent rules, resulting in \(-r^5 s^5 t^5\) since raising a negative number to an odd power keeps the negative sign. This simplified version is easier to work with for subsequent calculations or evaluations.

Powers of Variables

The concept of powers or exponents is an essential tool in algebra when dealing with variables. Exponents denote how many times a number, known as the base, is multiplied by itself. In our expression \((-r s t)^{5}\), we raise a product of variables to a power.
When you raise variables to a power, multiply the exponent by the power applied through **exponentiation rules**:

• If a variable is raised to a power, and that power is further raised to another power, you multiply the exponents together.
• Negative signs in front are evaluated based on whether the exponent is odd or even.

For variables like \(r\), \(s\), and \(t\) in our problem, \((-r s t)^{5}\) becomes \( r^5 \, s^5 \, t^5\), each letter gaining its own exponent. Through this distribution of power, each retains the base form, but the exponent indicates repeated multiplication by itself, fundamentally altering how the variable behaves in the expression.