Problem 75 - Algebra A Combined Function

Question

Use rational expressions to write as a single radical expression. $$ \frac{\sqrt[3]{b^{2}}}{\sqrt[4]{b}} $$

Step-by-Step Solution

Answer

The expression simplifies to \( \sqrt[12]{b^5} \).

Step 1: Express radicals as rational exponents

First, write the cube root and the fourth root with rational exponents. The cube root \( \sqrt[3]{b^2} \) becomes \( b^{2/3} \), and the fourth root \( \sqrt[4]{b} \) becomes \( b^{1/4} \).

Step 2: Create a single fraction with exponents

The expression \( \frac{\sqrt[3]{b^2}}{\sqrt[4]{b}} \) now becomes \( \frac{b^{2/3}}{b^{1/4}} \). Use the property of exponents \( \frac{a^m}{a^n} = a^{m-n} \) to combine the terms: \( b^{2/3 - 1/4} \).

Step 3: Simplify the exponents

Subtract \( \frac{1}{4} \) from \( \frac{2}{3} \). To do this, find a common denominator. The least common multiple of 3 and 4 is 12. Convert the fractions: \( \frac{2}{3} = \frac{8}{12} \) and \( \frac{1}{4} = \frac{3}{12} \). Hence, \( b^{2/3 - 1/4} = b^{8/12 - 3/12} = b^{5/12} \).

Step 4: Rewrite as a single radical

Convert the expression back to a radical form using the rational exponent as a root. \( b^{5/12} \) becomes \( \sqrt[12]{b^5} \).

Key Concepts

Radical Expressions

Radical expressions are math expressions that involve roots, such as square roots or cube roots. They can look a bit intimidating, but with some understanding, they're quite manageable. Radical expressions are often written using a radical symbol, like the square root sign \( \sqrt{} \).

For example, \( \sqrt{b} \) is a radical expression where \( b \) is the radicand. Radicals can have any degree like square root, cube root, fourth root, and so on. In our exercise, we're looking at cube and fourth roots.

To express cube roots and fourth roots as radicals, remember:

• Cube root is written as \( \sqrt[3]{b} \)
• Fourth root is written as \( \sqrt[4]{b} \)

It's essential to understand that radicals can be simplified or combined, especially when they have similar radicands, as seen in our example where the radicands are similar.

Exponents

Exponents are a shorthand way of expressing repeated multiplication. For example, \( b^3 \) means \( b \times b \times b \). In math, understanding exponents is crucial because they help simplify expressions and solve equations more easily. Exponents are significant in the world of algebra, calculus, and beyond.

They follow certain rules like:

• \( a^m \times a^n = a^{m+n} \)
• \( \frac{a^m}{a^n} = a^{m-n} \)
• \( (a^m)^n = a^{m \times n} \)

These rules are very helpful when working with rational expressions, simplifying them, or converting forms, as in our given problem. Connecting exponents and roots can transform complicated expressions into something much simpler to cope with.

Simplifying Fractions

In any math problem involving fractions, simplifying them often makes follow-up steps easier. Simplifying fractions with exponents isn't much different except you're working with the current exponent rules.

For instance, when we have a fraction like \( \frac{b^{2/3}}{b^{1/4}} \), we use the division rule \( \frac{a^m}{a^n} = a^{m-n} \). This rule allows us to combine exponents in fractions to a single expression, which makes the expression simpler and more straightforward.

In our case, simplifying \( \frac{b^{2/3}}{b^{1/4}} \) leads to \( b^{2/3 - 1/4} \). Obtaining a common denominator is a crucial step in subtraction (or addition) of fractions, just like you'd do with ordinary numbers.

Rational Exponents

Rational exponents offer a flexible way to write and handle root expressions. Instead of writing roots with a radical, we express them as fractional exponents.

For example, the nth root of \( a \) can be written as \( a^{1/n} \). It means the same thing and carries the same operations as rooting, but it is easier to use in algebraic manipulations. In our exercise:

• Cube root gives \( b^{2/3} \)
• Fourth root gives \( b^{1/4} \)

This transformation opens up the possibility to use exponent rules, like for multiplication or division. Finally, converting back from a rational exponent like \( b^{5/12} \) to the original radical form results in \( \sqrt[12]{b^5} \), bringing us back full circle between radicals and exponents.