Problem 48 - Algebra 2

Question

Write each expression in simplest form. Assume that all variables are positive. $$\left(\frac{x^{\frac{1}{4}}}{y^{-\frac{3}{4}}}\right)^{12}$$

Step-by-Step Solution

Answer

The simplified form of the expression is \(x^{3} \cdot y^{9}\)

Step 1: Apply the power rule

When raising one power to another power, we multiply the exponents. Therefore, \((x^{\frac{1}{4}})^{12}\) becomes \(x^{\frac{1}{4} \cdot 12}=x^{3}\). In the same way, \((y^{-\frac{3}{4}})^{12}\) becomes \(y^{-\frac{3}{4} \cdot 12}=y^{-9}\). This simplifies the original expression to \(\frac{x^{3}}{y^{-9}}\).

Step 2: Simplify negative exponents

Negative exponents are the reciprocals of the similar positive exponents. Therefore, \(\frac{x^{3}}{y^{-9}}\) simplifies to \(x^{3} \cdot y^{9}\), since the \(y^{-9}\) inside the denominator comes up to the numerator and becomes \(y^{9}\).

Step 3: Write in simplest form

The final simplified expression is \(x^{3} \cdot y^{9}\)

Key Concepts

Exponents

Exponents are a mathematical concept that allow us to express repeated multiplication of a number by itself. They are written as a small number, known as the exponent, placed to the upper right of a base number. For instance, in - \( x^3 \) the base is \( x \) and the exponent is \( 3 \). This means \( x \) is multiplied by itself three times: \( x \cdot x \cdot x \).Exponents are handy tools in both pure and applied mathematics. They help simplify expressions and make them easier to read and compute. This is particularly useful when dealing with large numbers or simplifying expressions in algebra. Understanding exponents is essential for mastering more advanced math topics as well.

Power Rule

The power rule is a fundamental property when working with exponents. It states that if you have a power raised to another power, you multiply the exponents. This is incredibly useful when working with expressions that require simplification, like raising a fraction or a product to a power.For example, - If you have \( (x^a)^b \), you apply the power rule by multiplying the exponents: \( x^{a\cdot b} \).In the given exercise:- \( (x^{\frac{1}{4}})^{12} \) becomes \( x^{\frac{1}{4} \, \cdot \, 12} = x^3 \).By reducing the complexity of working with nested exponents, the power rule makes algebraic manipulations more straightforward. It’s a vital skill to grab as you progress in mathematics.

Negative Exponents

Negative exponents might seem a bit counterintuitive at first, but they follow a straightforward rule. A negative exponent indicates a reciprocal. It means you take the inverse of the base raised to the absolute value of the exponent.Consider the expression:- \( x^{-a} \) becomes \( \frac{1}{x^a} \).This principle helps transform expressions containing negative exponents into more conventional forms without negatives. For example, - \( y^{-9} \) becomes \( \frac{1}{y^9} \).In the exercise solution:- The expression \( \frac{x^3}{y^{-9}} \) is simplified to \( x^3 \cdot y^9 \) by moving \( y^{-9} \) from the denominator to the numerator as a positive exponent. Understanding negative exponents is crucial for handling a variety of problems, especially in equations involving both multiplication and division of terms.

Simplification of Expressions

Simplifying expressions is a key skill in algebra. It involves reducing expressions to their simplest form, making them easier to work with. Simplification combines several algebraic rules and properties, including dealing with exponents, to present an expression in a more manageable way.When simplifying, follow these general steps:

• Apply any relevant rules, like the power rule, to combine or reduce exponents.
• Address any negative exponents by finding their reciprocals.
• Combine like terms and remove unnecessary parts of the expression.

In the exercise, - After applying the power rule and simplifying negative exponents, the expression \( \frac{x^3}{y^{-9}} \) is converted to its simplest form: \( x^3 \cdot y^{9} \).Simplification often involves small changes that lead to algebraic expressions that are far more efficient to work with, especially when solving equations or performing further calculations.