Problem 41 - Algebra 1: Concepts and Skills

Question

Simplify the variable expression. $$ \left(y^{1 / 6}\right)^{3} \cdot \sqrt{x} $$

Step-by-Step Solution

Answer

The simplified expression is \((yx)^{1/2}\).

Step 1: Simplify y's Exponent

An exponent of a power can be simplified by multiplying the exponents. For \(\left(y^{1 / 6}\right)^{3}\), we multiply \(1/6\) by 3 to get \(y^{1/2}\). The expression becomes \(y^{1/2} \cdot \sqrt{x}\).

Step 2: Convert Root to Exponential Form

\(\sqrt{x}\) is equivalent to \(x^{1/2}\). So the expression becomes \(y^{1/2} \cdot x^{1/2}\).

Step 3: Combine Like Terms

Expressions with the same exponent can be combined into a single term. Thus, \(y^{1/2} \cdot x^{1/2}\) can be written as \((yx)^{1/2}\). By doing so, the simplification is completed.

Key Concepts

Exponents

Exponents are a way to represent repeated multiplication of a number by itself. For instance, when you see something like \( a^n \), it means \( a \) is multiplied by itself \( n \) times. In this given problem, we start with understanding how to handle nested exponents in expressions like \( (y^{1/6})^3 \).

- **Multiplication of Exponents**: When you have an expression like \( (a^m)^n \), you can simplify it by multiplying the exponents, resulting in \( a^{m imes n} \). Here, \( (y^{1/6})^3 \) simplifies to \( y^{1/2} \) because \( 1/6 \times 3 = 1/2 \).
- **Interpreting Fractional Exponents**: These represent roots as well as powers. For example, \( a^{1/n} \) represents the \( n \)-th root of \( a \). When you see \( y^{1/2} \), it implies the square root of \( y \), which connects us to the next topic of radicals.

Grasping these exponent rules allows us to simplify complex expressions efficiently, which is especially useful in algebra and higher mathematics.

Radicals

Radicals are another way to denote roots, like square roots and cube roots. You often see them used with the radical sign (\( \sqrt{} \)). In our problem, we turn our attention to \( \sqrt{x} \).

- **Understanding \( \sqrt{x} \)**: The square root function \( \sqrt{x} \) is another representation of \( x^{1/2} \). Converting radicals to exponents is a helpful skill since exponent notation is often easier to manipulate algebraically.
- **Conversion and Simplification**: By rewriting \( \sqrt{x} \) as \( x^{1/2} \), we can handle radicals like other exponential expressions, allowing easier simplification.

Understanding radicals and how to convert them into exponents helps streamline the process of algebraic manipulation, smoothing the way for combining terms in various algebra problems.

Variable Expressions

Variable expressions are mathematical phrases that include numbers, variables (like \( x \) and \( y \)), and operation symbols. In our example, we are asked to simplify \( y^{1/2} \cdot x^{1/2} \).

- **Combining Like Terms**: When simplifying expressions with like bases and the same exponent, you can combine them into a single expression. Here, both \( y \) and \( x \) have the exponent \( 1/2 \). This means you can rewrite \( y^{1/2} \cdot x^{1/2} \) as \( (yx)^{1/2} \).
- **Simplification Techniques**: Always look for common bases and exponents. This will allow you to simplify the expressions further by factoring or regrouping.

Understanding how to handle variable expressions is crucial in algebra because it unfolds simpler paths to solving equations and understanding function behavior.