Problem 26 - Algebra 2 and Trigonometry

Question

Simplify each expression. In each exercise, all variables are positive. \(\frac{\left(a^{x}\right)^{y} b}{a^{x y}}\)

Step-by-Step Solution

Answer

The simplified expression is \( b \).

Step 1: Apply the Power Rule

The Power Rule in exponents states that \( (a^m)^n = a^{m \cdot n} \). Apply this to the numerator \( (a^x)^y \): \[(a^x)^y = a^{x \cdot y}\] Now the expression becomes \[\frac{a^{x \cdot y} \cdot b}{a^{x y}}\]

Step 2: Simplify Using the Quotient Rule

The Quotient Rule for exponents states that \( \frac{a^m}{a^n} = a^{m-n} \). Use this rule to simplify \( \frac{a^{x \cdot y}}{a^{x y}}\):\[\frac{a^{x \cdot y}}{a^{x y}} = a^{x \cdot y - x y}\] Since \( x \cdot y = x y\), the expression simplifies further:\[a^{0} = 1\] Therefore, now the expression becomes \( 1 \cdot b \) which is just \( b \).

Key Concepts

Power Rule

The Power Rule is a simple yet fundamental rule in dealing with exponents. It helps simplify expressions where a power itself is raised to another power. Imagine you have an expression like - \((a^m)^n\)Here's where the Power Rule comes into play. According to this rule, you can simplify this as - \(a^{m \cdot n}\)This means you're multiplying the exponents together to create a single exponent. This rule is extremely useful when you need to condense complex expressions into a smaller, more manageable form. This rule allows simplifying math expressions quickly and easily without expanding every step.

So, when you see a power raised to another power, remember you can use this rule to tidy your terms up neatly!

Quotient Rule

The Quotient Rule is another important exponent house rule you need in your toolkit. It simplifies expressions involving division of terms that contain the same base with different exponents. Here's what it looks like:- \(\frac{a^m}{a^n} = a^{m-n}\)This means you subtract the exponent in the denominator from the exponent in the numerator. This rule is all about balance - it eliminates unnecessary complexity and brings order to your algebraic expressions.

When faced with something like this in an expression, just remember to subtract the powers. Simplification becomes much smoother, helping you solve equations more efficiently.

The Quotient Rule is there when division and similar bases appear together; it helps secure that neat, solvable expression you're aiming for.

Simplification of Expressions

Simplifying expressions is all about reducing a complex equation into its simplest form. This makes it easier for you to work with it further or to identify the simplest answer. To simplify an expression means:

• Combine like terms
• Use rules, such as Power and Quotient Rule
• Cancel out terms

Your goal is to make the expression as neat and straightforward as possible without changing its value.

When simplifying, you start with components like variables and coefficients that can be combined, using algebraic rules.

This process lightens your work in methods like factorization, equation solving, and even when graphing functions.

Keeping the expression small translates to saving time and minimizing errors.

Algebraic Expressions

Algebraic expressions are made up of variables, constants, and operations. They form the backbone of algebra. It's the language that helps write complex real-world problems in a mathematical format. Here's what an algebraic expression can contain:

• Variables: like \(x\), \(y\), and \(a\)
• Constants: like numbers \(1, 2, 3\)
• Operations: such as addition \((+)\), subtraction \((-\)), multiplication \((* or \cdot)\), and division \((/ or \div)\)

These expressions can be as simple as a number or as complex as a polynomial. Simplifying them helps reveal their true value and can often resemble real-world situations like calculating interest or projectile paths.

Effective use of exponent rules and algebra rules makes handling algebraic expressions quicker and cleaner. You focus on making these expressions optimal so answers pop out effortlessly.