Problem 52 - Algebra for College Students

Question

Write each of the following using positive rational exponents. For example, \(\sqrt{a b}=(a b)^{\frac{1}{2}}=a^{\frac{1}{2}} b^{\frac{1}{2}}\). \(\sqrt[6]{a b^{5}}\)

Step-by-Step Solution

Answer

\(a^{\frac{1}{6}} b^{\frac{5}{6}}\)

Step 1: Identify and Rewrite the Radical Expression

The given expression is \(\sqrt[6]{a b^5}\), which denotes a sixth root. A root can be expressed as an exponent by writing the base raised to the reciprocal of the root. In this case, the sixth root will be expressed as an exponent of \(\frac{1}{6}\).

Step 2: Apply the Exponential Form to Each Factor

Write the expression under the radical as a product raised to the power of \(\frac{1}{6}\):\[(a b^5)^{\frac{1}{6}}\]

Step 3: Distribute the Exponent

Apply the exponent \(\frac{1}{6}\) to each factor inside the parentheses. This means raising both \(a\) and \(b^5\) to the power of \(\frac{1}{6}\): \[a^{\frac{1}{6}} (b^5)^{\frac{1}{6}}\]

Step 4: Simplify the Expression

Remember that when you have \((b^5)^{\frac{1}{6}}\), you use the power of a power property where you multiply exponents. Thus, it becomes: \[a^{\frac{1}{6}} b^{\frac{5}{6}}\]

Key Concepts

Radical Expressions

Radical expressions are quite frequent in algebra, representing the root of a number or expression. A simple radical such as \( \sqrt{a} \) represents the square root. The number 2 is implied as the index (root) when not written explicitly.

**Understanding Radicals**:

• A radical symbol \( \sqrt[n]{x} \) indicates the "n-th" root of \( x \).
• The number \( n \) is the "index", telling us which root we are taking. If no index is shown, it’s generally a square root (index = 2).

Radicals can seem tricky, but they simplify expressions in various mathematical problems. Knowing how to convert radicals into exponential form helps enormously. The shortcut? Remember that roots are just exponents written differently.
Examples make this clearer:

• \( \sqrt[3]{x} = x^{\frac{1}{3}} \) means the cube root of \( x \).
• \( \sqrt[6]{y^2} = y^{\frac{2}{6}} = y^{\frac{1}{3}} \).

When you see a radical expression, think about how you could express it with a rational exponent; it often simplifies your calculations.

Exponents

Exponents are a way to represent repeated multiplication concisely. If you have \( a^n \), \( a \) is the base and \( n \) is the exponent, meaning "multiply \( a \) by itself \( n \) times".

**Key Points about Exponents**:

• An exponent of 1 means the number itself (\( a^1 = a \)).
• A zero exponent means one (\( a^0 = 1 \)), as long as \( a eq 0 \).
• Negative exponents represent the reciprocal (\( a^{-n} = \frac{1}{a^n} \)).

When working with rational exponents, which are just fractions, they represent both roots and powers:

• \( a^{\frac{1}{n}} \) means the n-th root of a.
• \( a^{\frac{m}{n}} = (\sqrt[n]{a})^m = \sqrt[n]{a^m} \).

It’s useful to remember the power rules, like \((a^m)^{n} = a^{m\times n}\), which simplify interacting expressions with more than one exponent.
This helps when simplifying expressions like \((b^5)^{\frac{1}{6}}\) to \(b^{\frac{5}{6}}\).

Algebraic Simplification

Algebraic simplification involves reducing expressions to their simplest form. The goal is to make equations easier to understand and solve.

**Steps of Simplification**:

• Combine like terms - Terms that have identical variable parts.
• Use distributive properties to remove parentheses, where possible.
• Simplify any fractionals or radical parts by expressing them in simpler exponential forms.

Radicals, especially, become less complex when expressed using rational exponents. This is pivotal, as demonstrated in the simplification from \( (a b^5)^{\frac{1}{6}} \) to \( a^{\frac{1}{6}} b^{\frac{5}{6}} \). By breaking down this step:

• Apply the fractional exponent \( \frac{1}{6} \) to both \( a \) and \( b^5 \), individually.
• Leverage properties of exponents, like \((b^5)^{\frac{1}{6}} = b^{5 \times \frac{1}{6}} = b^{\frac{5}{6}}\).

By understanding these properties, you free up mental space by solving complex expressions element by element, keeping analysis systematic and clear.