Problem 98 - Intermediate Algebra

Question

Use rational expressions to write as a single radical expression. $$ \sqrt[3]{b} \cdot \sqrt[5]{4 a} $$

Step-by-Step Solution

Answer

The single radical expression is \(\sqrt[15]{b^5 \cdot 4^3 \cdot a^3}\).

Step 1: Rewrite as Exponents

Begin by expressing each radical using fractional exponents. Recall that \(\sqrt[n]{x} = x^{1/n}\). This gives us \(b^{1/3}\) for \(\sqrt[3]{b}\) and \((4a)^{1/5}\) for \(\sqrt[5]{4a}\).

Step 2: Multiply the Exponents

Multiply the two expressions with fractional exponents: \(b^{1/3} \cdot (4a)^{1/5} = b^{1/3} \cdot 4^{1/5} \cdot a^{1/5}\). Separate \((4a)^{1/5}\) into \(4^{1/5} \cdot a^{1/5}\).

Step 3: Combine into Single Radical

To combine into a single radical, use the properties of exponents to find a common exponent base. This can be written as a single radical: \( (b^{1/3} \cdot 4^{1/5} \cdot a^{1/5})\).

Step 4: Simplify as a Single Radical

Since the expressions have different bases, you combine them under a single expression, which can be expressed as \(\sqrt[15]{b^5 \cdot 4^3 \cdot a^3}\). The least common multiple of the denominators in the exponents (3 and 5) is 15. Now express each term with this common denominator exponent.

Step 5: Final Expression

Combine all terms under the new radical: \(\sqrt[15]{b^5 \cdot 4^3 \cdot a^3}\), by raising each component to the power needed to match a common denominator of 15. Therefore, \(b^{1/3} = b^{5/15}\), \(4^{1/5} = 4^{3/15}\), and \(a^{1/5} = a^{3/15}\).

Key Concepts

Fractional Exponents

Fractional exponents are a way to express radicals in exponential form, which can make complex calculations easier. Instead of writing radicals like \(\sqrt[n]{x}\), we express them as \(x^{1/n}\). This approach creates a versatile way to handle expressions that include both roots and powers.

• If you have \(\sqrt[3]{b}\), it is equivalent to \(b^{1/3}\).
• For \(\sqrt[5]{4a}\), you write it as \((4a)^{1/5}\).

Using fractional exponents is beneficial because it allows direct use of the same rules that apply to mathematical operations involving integers, thus simplifying multiplication, division, and other operations in algebra.New lines (
) every 2-3 sentences help with readability.

Multiplication of Radicals

When multiplying radicals, using fractional exponents is very effective. This allows you to apply familiar rules of exponents and make calculations less cumbersome. Consider the expression \(b^{1/3} \cdot (4a)^{1/5}\). By expressing radicals with exponents, we can decompose the expression further as \(b^{1/3} \cdot 4^{1/5} \cdot a^{1/5}\).

• The key here is to treat each component independently and apply the laws of exponents.
• The multiplication process becomes just a straightforward application of exponent rules.

Using this approach, breaking down the expression into smaller parts helps grasp the entire multiplication process more clearly.Remember that keeping expressions simple leads to fewer errors.

Simplifying Radicals

Simplifying radicals is a crucial step in many algebraic operations. Our initial expression: \(b^{1/3} \cdot 4^{1/5} \cdot a^{1/5}\) can be combined and simplified into a single radical form.

• Begin by finding a common denominator for the exponents. This helps align all parts of the expression.
• The least common multiple (LCM) here is 15, which dictates the root we will use.

Upon simplifying, you come to the expression \(\sqrt[15]{b^5 \cdot 4^3 \cdot a^3}\). Recognizing the need for a common root makes simplifying radicals far more consistent and systematic.Focus on one term at a time to prevent overcomplication.

Least Common Multiple of Exponents

The least common multiple (LCM) of exponents is pivotal when consolidating expressions into a single term. For our task, we needed to find the LCM of 3 and 5 (the denominators in \(b^{1/3}\) and \((4a)^{1/5}\)).

• The LCM of two numbers is the smallest number evenly divisible by both. Here, it is 15.
• Utilizing the LCM, we transform \(b^{1/3}\) to \(b^{5/15}\), \(4^{1/5}\) to \(4^{3/15}\), and \(a^{1/5}\) to \(a^{3/15}\).

This common base now allows all parts to be integrated under one root, \(\sqrt[15]{b^5 \cdot 4^3 \cdot a^3}\). Having a shared base simplifies mathematical manipulation by standardizing the expression.