Problem 19 - Algebra A Combined Function

Question

Use the quotient rule to simplify. Assume that all variables represent positive real numbers. $$ \sqrt[3]{\frac{4}{27}} $$

Step-by-Step Solution

Answer

The expression simplifies to \( \frac{\sqrt[3]{4}}{3} \).

Step 1: Express the Radical as a Fractional Exponent

The expression \( \sqrt[3]{\frac{4}{27}} \) can be rewritten using fractional exponents. The cube root of a fraction \( \frac{a}{b} \) is \( (\frac{a}{b})^{1/3} \). So we have: \[ \left( \frac{4}{27} \right)^{1/3} \]

Step 2: Apply the Quotient Rule

According to the rule of exponents for quotients, \( \left( \frac{a}{b} \right)^n = \frac{a^n}{b^n} \). Applying this to \( \left( \frac{4}{27} \right)^{1/3} \), we get: \[ \frac{4^{1/3}}{27^{1/3}} \]

Step 3: Simplify the Numerator and Denominator

We simplify each part separately. The cube root of 4 is \( 4^{1/3} \), and the cube root of 27 is \( 27^{1/3} \). Since 27 is \( 3^3 \), its cube root is 3: \[ \frac{4^{1/3}}{3} \]

Step 4: Express the Simplified Expression

The simplified form of the original expression is \( \frac{\sqrt[3]{4}}{3} \). There is no further simplification since \( \sqrt[3]{4} \) does not resolve neatly to a rational number.

Key Concepts

Fractional Exponents

Fractional exponents are a way of expressing powers and roots together in one concise form. For instance, the expression \( x^{1/n} \) is equivalent to the \( n \)-th root of \( x \). This means if you see \( x^{1/3} \), it translates to the cube root of \( x \). The beauty of fractional exponents lies in their ability to simplify complex radical expressions, making calculations and manipulations more straightforward.

You can think of fractional exponents as a method of breaking down roots. For example:

• The square root of \( a \) can be expressed as \( a^{1/2} \).
• The cube root of \( a \) becomes \( a^{1/3} \).

Fractional exponents follow the same laws as integer exponents, which means you can apply rules like the power rule and the quotient rule. This makes them extremely useful for simplifying expressions.

Simplifying Radicals

Simplifying radicals involves expressing a radical expression in its simplest form. When you encounter a radical, such as a square root or cube root, you aim to reduce it to the simplest radical terms possible. This often involves converting the expression into fractional exponents.

The process typically includes a few key steps:

• Convert the radical to an exponent, for example, the cube root of \( \frac{4}{27} \) becomes \( (\frac{4}{27})^{1/3} \).
• Apply the exponent rules, such as the quotient rule, to break the expression down into more manageable parts.
• Simplify each part separately. For instance, compute \( 27^{1/3} \) which simplifies to 3 since \( 27 = 3^3 \).

Once all parts are simplified, combine them back into a single expression. Sometimes, as with \( \frac{\sqrt[3]{4}}{3} \), it cannot be broken down further and has to be left in its simplest radical form.

Positive Real Numbers

When working with radicals and fractional exponents, it's important to assume variables represent positive real numbers. Positive real numbers are all the numbers greater than zero that do not have an imaginary component. This assumption simplifies processes because you avoid dealing with complex numbers.

By sticking to positive real numbers:

• Radicals and fractional exponents have clear, real values.
• You avoid undefined situations that occur with negative bases and non-integer exponents, like \( (-3)^{1/2} \) which doesn't have a real number solution.

Understanding that variables are positive ensures calculations follow predictable and logical paths, making the simplification process more straightforward.