Problem 43 - Algebra A Combined Function

Question

Use the quotient rule to simplify. Assume that all variables represent positive real numbers. $$ \sqrt[4]{a^{8} b^{7}} $$

Step-by-Step Solution

Answer

The simplified expression is \( a^{2} b^{7/4} \).

Step 1: Understand the Problem

We need to simplify the expression \( \sqrt[4]{a^{8} b^{7}} \) using the properties of exponents and roots. The expression is under a fourth root.

Step 2: Use the Property of Roots

The expression \( \sqrt[4]{a^{8} b^{7}} \) can be rewritten using the property of roots as \( (a^{8} b^{7})^{1/4} \). This means we will raise the entire expression inside the parenthesis to the power of \( \frac{1}{4} \).

Step 3: Apply the Power to Each Factor

Apply the exponent \( \frac{1}{4} \) to each component inside the expression separately: \( a^{8 \times \frac{1}{4}} \) and \( b^{7 \times \frac{1}{4}} \).

Step 4: Simplify Each Component

Calculate each exponent individually: \( a^{8 \times \frac{1}{4}} = a^{2} \) and \( b^{7 \times \frac{1}{4}} = b^{7/4} \).

Step 5: Combine the Results

Combine the simplifications of each factor: \( a^{2} b^{7/4} \). Now the expression is fully simplified.

Key Concepts

Properties of Exponents

The properties of exponents are essential tools in algebra that allow us to manipulate expressions involving powers, making them simpler to work with. Understanding these properties enables you to rewrite expressions in different forms without changing their value.

Here are a few key properties:

• **Product of Powers Property**: When multiplying like bases, add the exponents: \[ x^m \cdot x^n = x^{m+n} \].
• **Power of a Power Property**: When raising a power to another power, multiply the exponents: \[ (x^m)^n = x^{m \cdot n} \].
• **Power of a Product Property**: Distribute the exponent over a product: \[ (xy)^m = x^m \cdot y^m \].
• **Quotient of Powers Property**: When dividing like bases, subtract the exponents: \[ \frac{x^m}{x^n} = x^{m-n} \], where \( x eq 0 \).

In the original problem, these properties were employed to simplify the expression under the fourth root. By applying these rules, each part of the expression could be managed separately before combining the results for the final answer.

Fourth Roots

Roots are special mathematical operations that are the inverse of exponents. The fourth root is simply one specific type of root. It asks what number, when multiplied by itself four times, gives the original number.

For example, the fourth root of 16 is 2 because \(2^4 = 16\).

When we see an expression like \( \sqrt[4]{a^8} \), we're looking for a number that, when raised to the fourth power, equals \(a^8\).To simplify such expressions, the property of **Power of Roots** is useful, which is \( \sqrt[n]{x} = x^{1/n} \). This property helps turn roots into exponents, facilitating further simplification using exponent rules. In the exercise, applying the fourth root means rewriting \( \sqrt[4]{a^8b^7} \) as \((a^8b^7)^{1/4}\), making it easier to handle using the rules of exponents.

Quotient Rule

The quotient rule is a fundamental principle used in algebra and calculus to handle expressions involving division, particularly when they are in exponent form.

In simpler terms, the quotient rule states that for any non-zero number, you can manage the division of powers with the same base as follows: \[ \frac{a^m}{a^n} = a^{m-n} \].

This rule allows you to simplify expressions by subtracting the exponent of the denominator from the exponent of the numerator if the bases are the same.

While the original exercise doesn't explicitly show division, understanding how to manage exponents through division supports overall comprehension of expression simplification. When faced with complex expressions, breaking them down into manageable parts using these types of rules makes solving them more straightforward. Applying the principles of exponents, even implicitly, demonstrates the power of knowing these rules for effective algebraic manipulation.