Problem 73 - Intermediate Algebra

Question

Use the quotient rule to divide. Then simplify if possible. See Example 5 $$ \frac{\sqrt[4]{96 a^{10} b^{3}}}{\sqrt[4]{3 a^{2} b^{3}}} $$

Step-by-Step Solution

Answer

The simplified expression is \(2a^2\).

Step 1: Apply the Quotient Rule for Radicals

When you have two radical expressions with the same index, you can divide the radicands directly. Here, the index is 4 (fourth root). Apply the quotient rule: \[\frac{\sqrt[4]{96 a^{10} b^{3}}}{\sqrt[4]{3 a^{2} b^{3}}} = \sqrt[4]{\frac{96 a^{10} b^{3}}{3 a^{2} b^{3}}}\]

Step 2: Simplify the Radicand

Divide the terms under the radical. First, simplify the coefficients: \[\frac{96}{3} = 32\]For the variables, apply the quotient rule for exponents: \[(a^{10} \div a^{2}) = a^{8}\], and \[(b^{3} \div b^{3}) = b^{0} = 1\].Thus, the expression simplifies to:\[\sqrt[4]{32 a^{8}}\]

Step 3: Simplify the Fourth Root

Now simplify the fourth root: - For \(32\), recall that \(32 = 2^5\). Since we need pairs of four for the fourth root, it contributes \(2\) out of \(2^4 = 16\). - \[a^{8}\] becomes \[(a^2)^4\], which results in \(a^2\) when taking the fourth root.Combine these: \[\sqrt[4]{32 a^{8}} = 2a^2\]

Step 4: Verify and Present the Final Expression

After evaluating the steps, the simplified form of the given expression is:\[2a^2\]

Key Concepts

Simplifying Radicals

Simplifying radicals involves reducing the expression inside the radical to its simplest form. In our problem, we start with a radical expression that includes numbers and variables, all under the fourth root.
To simplify means to:

• Break down numbers to their prime factors.
• Apply exponent rules to simplify powers of variables.
• Use \[\quad\] formula.\[\sqrt[n]{a} \cdot \sqrt[n]{b} = \sqrt[n]{a \cdot b}\]

In our solution, dividing and removing common factors inside the radical are key steps to making the expression simpler. Remember, when terms within a radical share a common factor, simplify them before moving on to the root extraction steps.

Radical Expression

A radical expression is any mathematical expression that contains a radical symbol, which looks like this: \(\sqrt{}\). Radicals can have various indices like square roots \(\sqrt{}\) (index 2), cube roots \(\sqrt[3]{}\) (index 3), and more.

• Here, we are working with the fourth root, hence the radical is \(\sqrt[4]{}\).
• It's important to remember that the expression inside the radical is called the radicand.
• Often, radical expressions contain variables, coefficients, and exponents.

In our exercise, we simplified the radical expression by first considering the radicands \(96 a^{10} b^3\) and \(3 a^2 b^3\). Understanding the structure of the radical is crucial before tackling simplification.

Fourth Root

The fourth root of a number or expression refers to a value that, when raised to the power of four, equals the original number. Mathematically, it is represented as \(\sqrt[4]{}\).

• Finding the fourth root involves determining which quantity needs to be multiplied by itself four times to get the number inside the radical.
• For example, the fourth root of \((2^4)\) is 2, since \(2 \times 2 \times 2 \times 2 = 16\).

In the exercise, we calculated the fourth root of \(32 a^8\). Using prior knowledge, we broke \(32\) into prime factors and noted that \(a^8\) simplifies to \(a^2\) when processed through the fourth root, illustrating the concept clearly.

Quotient Rule for Exponents

The quotient rule for exponents is applicable when dividing expressions that have the same base. It states that when you divide like bases, you subtract the exponents: \[\frac{a^m}{a^n} = a^{m-n}\]This rule simplifies calculations greatly and is essential for reducing complex expressions.

• The quotient rule helped us reduce \(a^{10}\) divided by \(a^2\) to \(a^8\).
• It also simplified \(b^3\) divided by \(b^3\) to \(b^0 = 1\).

Remember, applying this rule is fundamental when radicals include variables with exponents. It ensures expressions are neat and manageable before tackling the root itself.