Problem 79 - Algebra A Combined Function

Question

Use rational expressions to write as a single radical expression. $$ \frac{\sqrt[3]{a^{2}}}{\sqrt[6]{a}} $$

Step-by-Step Solution

Answer

The single radical expression is \( \sqrt{a} \).

Step 1: Convert Radicals to Exponents

The expression contains radicals \(\sqrt[3]{a^2}\) and \(\sqrt[6]{a}\). To start, convert these into exponent form: \( a^{2/3} \) and \( a^{1/6} \) respectively. This uses the property \( \sqrt[n]{x^m} = x^{m/n} \).

Step 2: Simplify the Expression

Now rewrite the original expression using these exponents: \( \frac{a^{2/3}}{a^{1/6}} \). The next step involves subtracting the exponents, following the property \( \frac{x^m}{x^n} = x^{m-n} \).

Step 3: Subtract Exponents

Subtract the exponents: \( \frac{2}{3} - \frac{1}{6} \). First, convert \( \frac{2}{3} \) into \( \frac{4}{6} \). Then, \( \frac{4}{6} - \frac{1}{6} = \frac{3}{6} = \frac{1}{2} \).

Step 4: Convert the Result Back to a Radical

With the subtraction complete, the simplified expression is \( a^{1/2} \). Convert this back into a radical: \( \sqrt{a} \). This is the single radical expression of the original problem.

Key Concepts

Radical Expressions

Radical expressions are an integral part of algebra, often seen as the "roots" of numbers or variables. They involve the symbol \( \sqrt{} \) or its variations, such as cube roots \( \sqrt[3]{} \), representing the inverse operation of raising a number to a power.
For instance, \( \sqrt[3]{a^2} \) is the cube root of \( a^2 \), meaning we need a number that, when cubed, gives \( a^2 \). Transitioning between radicals and exponents helps simplify problems and is essential in algebra.
Useful properties to remember include:

• \( \sqrt[n]{x^m} = x^{m/n} \)
• \( \sqrt{x} = x^{1/2} \)
• \( \sqrt[3]{x} = x^{1/3} \)

This conversion is fundamental when handling complex expressions, allowing for more straightforward arithmetic operations.

Exponents

Exponents are a way to express repeated multiplication of a number by itself. In algebra, they are written as \( x^m \), meaning \( x \) is multiplied by itself \( m \) times. They make handling large numbers simpler.
Understanding the laws of exponents is crucial as they enable us to manipulate and simplify algebraic expressions:

• Product of Powers: \( x^m \cdot x^n = x^{m+n} \)
• Quotient of Powers: \( \frac{x^m}{x^n} = x^{m-n} \)
• Power of a Power: \( (x^m)^n = x^{m \cdot n} \)

In our task, converting \( \sqrt[3]{a^2} \) and \( \sqrt[6]{a} \) into exponents \( a^{2/3} \) and \( a^{1/6} \) lets us use these rules to simplify the expressions.

Simplifying Expressions

Simplifying expressions is the process of rewriting algebraic statements in a more concise form. This makes them easier to interpret and solve.
The process often involves:

• Converting radical expressions to exponents
• Using exponent rules to combine or simplify terms
• Reducing fractions or expressions by finding a common base

For example, the expression \( \frac{a^{2/3}}{a^{1/6}} \) can be simplified by subtracting the exponents, as per the quotient of powers property. This results in \( a^{1/2} \), which is simpler and represents the same value as the original expression.

Mathematical Properties

Mathematical properties are the rules that guide how numbers and expressions behave during arithmetic operations. Knowing these properties helps in problem-solving, allowing us to rearrange and simplify expressions logically.
Key properties include:

• Commutative Property: \( a + b = b + a \) or \( a \cdot b = b \cdot a \)
• Associative Property: \( (a + b) + c = a + (b + c) \) or \( (a \cdot b) \cdot c = a \cdot (b \cdot c) \)
• Distributive Property: \( a(b + c) = ab + ac \)
• Exponent Rules: including the aforementioned rules for handling powers

These principles are consistently used to convert, manipulate, and simplify expressions, ultimately activating the "math magic" that turns complex problems into simpler solutions.