Problem 28 - College Algebra with Modeling and Visualization

Question

Simplify the expression. Assume that all variables are positive. $$ \sqrt{6 x^{5}} \cdot \sqrt{6 x} $$

Step-by-Step Solution

Answer

The simplified expression is \(6x^3\).

Step 1: Rewrite the Radicals as Exponents

The expression involves square roots, which can be rewritten using exponent rules. Recall that \( \sqrt{a} = a^{1/2} \). So, we have \( \sqrt{6x^5} = (6x^5)^{1/2} \) and \( \sqrt{6x} = (6x)^{1/2} \). Thus, the expression becomes \( (6x^5)^{1/2} \cdot (6x)^{1/2} \).

Step 2: Apply the Product Rule for Exponents

The product of two powers with the same base can be written as a single power: \( a^{m} \times a^{n} = a^{m+n} \). Apply this to our expression: \((6x^5)^{1/2} \cdot (6x)^{1/2} = (6^1 x^5)^{1/2} \cdot (6^1 x^1)^{1/2} \) becomes \( ((6^1 x^5) \times (6^1 x^1))^{1/2} \).

Step 3: Combine the Exponents and Simplify

First, multiply the terms inside the radical: \( 6^1 \times 6^1 = 6^2 \) and \( x^5 \times x^1 = x^{5+1} = x^6 \). This gives us \( (6^2 x^6)^{1/2} \).

Step 4: Simplify the Radical Expression

Now, simplify the new expression: \( (6^2 x^6)^{1/2} \) can be calculated as \( 6^{2/2} \cdot x^{6/2} \). Since \( 2/2 = 1 \) and \( 6/2 = 3 \), this simplifies to \( 6^1 \cdot x^3 \), which is \( 6x^3 \).

Key Concepts

Radicals and Exponents

Understanding both radicals and exponents helps simplify expressions involving these components. A radical, specifically a square root, is just another way to represent a fractional exponent. To clarify, the square root of any number 'a' is written as \( \sqrt{a} = a^{1/2} \). This means when you see a square root, you can replace it with the exponent \( 1/2 \).
For example:

• \( \sqrt{6x^5} \) becomes \( (6x^5)^{1/2} \)
• Similarly, \( \sqrt{6x} \) is \( (6x)^{1/2} \)

Switching between radicals and exponents is crucial for simplifying expressions involving them. Recognizing this equivalence opens up possibilities for using properties of exponents to further modify and streamline your calculations.

Product Rule for Exponents

The product rule for exponents is an essential tool when working with expressions that have the same base. The rule states: if you multiply two powers with the same base, you can add their exponents. Mathematically, this is written as \( a^m \times a^n = a^{m+n} \).
When simplifying expressions like \( \sqrt{6x^5} \cdot \sqrt{6x} \), rewriting them as exponents allows you to use this rule:

• Convert to \( (6x^5)^{1/2} \cdot (6x)^{1/2} \)
• Recognize it as \( ((6^1 x^5) \times (6^1 x^1))^{1/2} \)

By applying the product rule, you combine \( 6^1 \times 6^1 \) which results in \( 6^2 \) and \( x^5 \times x^1 = x^{5+1} = x^6 \). This combination reduces the expression to \( (6^2 x^6)^{1/2} \), making it easier to handle in further steps.

Combining Like Terms

After applying the product rule for exponents and simplifying expressions, you typically need to combine like terms. In expressions that involve multiplication of similar bases, this means combining their exponents or coefficients.

• To evaluate \( (6^2 x^6)^{1/2} \), first simplify each factor separately.
• The term \( 6^2 \) under the square root simplifies to \( 6^{2/2} = 6^1 \) or just 6 because \( 2/2 \) equals 1.
• Similarly, \( x^6 \) becomes \( x^{6/2} \) or \( x^3 \) using the same principle.

Finally, combining these like terms finishes the simplification process, giving you a clean and compact form of the original expression: \( 6x^3 \).
By practicing these steps, you'll become more efficient at simplifying even the most complex expressions.