Problem 51 - Algebra 2 and Trigonometry

Question

In \(38-57,\) write each radical expression as a power with positive exponents and express the answer in simplest form. The variables are positive numbers. $$ \sqrt{\frac{3 a}{4 b}} $$

Step-by-Step Solution

Answer

The expression simplifies to \( \frac{\sqrt{3} \cdot a^{1/2}}{2 \cdot b^{1/2}} \).

Step 1: Identify the Expression

The given expression is a square root: \( \sqrt{\frac{3a}{4b}} \). This expression must be rewritten using positive exponents.

Step 2: Convert to Exponential Form

Recall that \( \sqrt{x} \) is equivalent to \( x^{1/2} \). Therefore, we can write \( \sqrt{\frac{3a}{4b}} \) as \( \left( \frac{3a}{4b} \right)^{1/2} \).

Step 3: Apply Power to Both Numerator and Denominator

Use the property \( \left( \frac{a}{b} \right)^{m} = \frac{a^m}{b^m} \) to apply the exponent \( \frac{1}{2} \) to both the numerator and the denominator separately, resulting in \( \frac{(3a)^{1/2}}{(4b)^{1/2}} \).

Step 4: Simplify the Powers

Apply the exponent to each part of the fraction: \( \frac{(3)^{1/2} (a)^{1/2}}{(4)^{1/2} (b)^{1/2}} \). Recognize \( 4^{1/2} = 2 \), so the denominator becomes \( 2(b^{1/2}) \).

Step 5: Write the Final Expression

Combine the results: \( \frac{\sqrt{3} \cdot a^{1/2}}{2 \cdot b^{1/2}} \). This is the simplest form of the expression using positive exponents.

Key Concepts

Radical Expressions

Radical expressions are mathematical phrases that include a root symbol, like a square root (\( \sqrt{} \)), cube root, or any other root. The number inside the root symbol, known as the radicand, is the value from which you are taking the root.
For example, in the expression \( \sqrt{9} \), the number 9 is the radicand, and since it is under a square root, you are essentially looking for a number which when multiplied by itself results in 9.
Radical expressions can sometimes seem complex, but they often have an equivalent expression that uses exponents, which can be easier to work with in algebraic manipulations. Understanding how to convert between radical and exponential notation is crucial, as it is an essential tool for simplifying expressions and solving equations.

Simplifying Expressions

Simplifying expressions refers to reducing the expression to its most concise and manageable form. When dealing with radical expressions, this often involves rewriting these expressions using positive exponents.
This simplification process can make complex expressions easier to understand and use in calculations. Let's consider our original expression \( \sqrt{\frac{3a}{4b}} \). By converting the radical into an expression with an exponent, it becomes \( \left( \frac{3a}{4b} \right)^{1/2} \).

• Use the power rule \( \left( \frac{a}{b} \right)^{m} = \frac{a^m}{b^m} \) to apply the exponent separately to the numerator and the denominator.
• Then, further simplify each component as needed. For instance, \( 4^{1/2} = 2 \).

This method maintains the integrity of the mathematical expression while making it cleaner and often easier to handle in further algebraic procedures.

Positive Exponents

Positive exponents indicate how many times a base number is used as a factor in a multiplication. This concept is fundamental when converting radical expressions into exponential form.
For instance, consider the square root \( \sqrt{x} \), which can be rewritten as \( x^{1/2} \). Here, the exponent \( 1/2 \) is positive, and this notation is essential in expressing radicals with positive exponents.
This transformation helps simplify expressions, making them easier to calculate and understand, as seen in \( \sqrt{\frac{3a}{4b}} = \left( \frac{3a}{4b} \right)^{1/2} \). Utilizing positive exponents allows these expressions to blend seamlessly with other algebraic expressions and be easily manipulated within equations.
Understanding and using positive exponents is especially beneficial in calculus and higher-level mathematics, where expressing ideas in their simplest exponential form can significantly ease solving complex problems.