Problem 39 - Algebra A Combined Function

Question

Use the quotient rule to simplify. Assume that all variables represent positive real numbers. $$ \sqrt{100 x^{5}} $$

Step-by-Step Solution

Answer

The expression simplifies to \( 10x^{5/2} \).

Step 1: Analyze the Expression

The expression given is \( \sqrt{100 x^5} \). This can be interpreted as the square root function applied to the product of a constant and a variable to the power of 5.

Step 2: Break Down the Square Root

The square root of a product, \( \sqrt{100 x^5} \), can be separated into the product of square roots: \( \sqrt{100} \times \sqrt{x^5} \).

Step 3: Simplify the Square Root of the Constant

Calculate \( \sqrt{100} \). The square root of 100 is 10 because 10 squared is 100. Thus, \( \sqrt{100} = 10 \).

Step 4: Simplify the Square Root of the Variable Expression

The expression \( \sqrt{x^5} \) can be written as \( x^{5/2} \). In general, \( \sqrt{x^n} = x^{n/2} \).

Step 5: Combine the Simplified Parts

Now, combine the results of the square roots: \( 10 \times x^{5/2} \). Thus, \( \sqrt{100x^5} = 10x^{5/2} \).

Key Concepts

Quotient Rule

The quotient rule is a fundamental tool in calculus, primarily used for differentiating functions that are the quotient of two other functions. However, when we talk about simplifying radicals, it's all about recognizing how we can "distribute" the square root across division. Let's say we have a fraction inside a square root. According to the quotient rule for radicals, we can simplify \( \sqrt{\frac{a}{b}} \) by rewriting it as \( \frac{\sqrt{a}}{\sqrt{b}} \), assuming \( b eq 0\). This means we can "break apart" the square root over a fraction into individual square roots of the numerator and the denominator.

• This is useful because it lets us handle each part separately, often simplifying our expressions quickly.
• Understanding this rule is key for making radical expressions more manageable, especially when dealing with complicated or large expressions.

In our exercise, while the quotient rule wasn't directly applied in a fraction format, the understanding that a square root over a product can be separated similarly helps in simplifying such expressions. Keep this rule handy as it's quite a powerful tool in the simplification toolkit.

Square Roots

Square roots appear everywhere in algebra and mathematics. A square root of a number or an expression is simply a value that, when multiplied by itself, recreates the original number or expression. When dealing with radicals, especially in simplification processes, understanding how to manipulate square roots is vital.

• The square root of a perfect square (like 100 or 81) is straightforward. For instance, \( \sqrt{100} = 10 \) because \(10^2 = 100\).
• If the expression inside the root isn't a perfect square, we often use factorization or exponent rules to simplify.

In our problem, we see this when \( \sqrt{100} \) was calculated directly, while \( \sqrt{x^5} \) required further manipulation using exponents. Becoming comfortable with finding square roots, especially separating components using property \( \sqrt{a\times b} = \sqrt{a} \times \sqrt{b} \), expands the ways these problems can be approached. Squares and roots are fundamental and frequently paired with exponents in algebraic calculus, making it crucial to grasp their properties.

Exponent Rules

Exponent rules are the foundation for simplifying expressions involving powers. When dealing with variables raised to powers, these rules offer a plethora of ways to rewrite and simplify expressions.

• One of the core rules is that multiplying like bases with exponents adds their powers, \(a^m \times a^n = a^{m+n}\).
• Another rule states that a power raised to another power multiplies the exponents, \((a^m)^n = a^{m\times n}\).

In simplifying radicals like \( \sqrt{x^5} \), we employ the rule converting roots to fractional exponents: \( \sqrt{x^5} = x^{5/2} \). This conversion is pivotal because it transforms the problem into a familiar algebraic expression involving exponents, which can be manipulated with standard rules. Understanding and using fractional exponents can transform complex radical problems into manageable equations. Mastery of these exponent rules not only aids in simplifying expressions but also in effectively solving more complex algebraic equations later on.