Problem 48 - Algebra 2 and Trigonometry

Question

In \(38-57,\) write each radical expression as a power with positive exponents and express the answer in simplest form. The variables are positive numbers. $$ \sqrt{64 a^{3} b^{6}} $$

Step-by-Step Solution

Answer

The simplest form is \( 8 \cdot a^{\frac{3}{2}} \cdot b^3 \).

Step 1: Understanding the Radicand

The expression given is \( \sqrt{64 a^3 b^6} \). Here, \(64\) and variables \(a\) and \(b\) are under the square root. We are looking to express this entire radical as a power with positive exponents.

Step 2: Express the Square Root as a Fractional Exponent

A square root suggests an exponent of \( \frac{1}{2} \). Hence, we rewrite \( \sqrt{64 a^3 b^6} \) as \( (64 a^3 b^6)^{\frac{1}{2}} \).

Step 3: Apply the Power of a Product Rule

The power of a product rule states that \( (xy)^n = x^n y^n \). Applying this to our expression, we have:\[ 64^{\frac{1}{2}} \cdot (a^3)^{\frac{1}{2}} \cdot (b^6)^{\frac{1}{2}} \].

Step 4: Simplify Each Component

Now, compute each part separately:- \( 64^{\frac{1}{2}} = 8 \) because \( 8^2 = 64 \).- \( (a^3)^{\frac{1}{2}} = a^{\frac{3}{2}} \) by multiplying the exponents.- \( (b^6)^{\frac{1}{2}} = b^3 \) because \( 6 \cdot \frac{1}{2} = 3 \).

Step 5: Write the Final Expression

Combine the simplified components to present the final expression:- \( 8 \cdot a^{\frac{3}{2}} \cdot b^3 \).

Key Concepts

Fractional Exponents

Converting roots into fractional exponents is a key concept in algebra that helps simplify complex expressions. When you encounter a square root, it can be expressed with an exponent of \( \frac{1}{2} \). For example, \( \sqrt{x} = x^{\frac{1}{2}} \). This transformation is just part of a broader rule where the \( n \)-th root of a variable or number can be expressed as a fractional exponent, \( x^{\frac{1}{n}} \).

• Cubic roots: \( \sqrt[3]{x} = x^{\frac{1}{3}} \)
• Fourth roots: \( \sqrt[4]{x} = x^{\frac{1}{4}} \)

In the provided expression, \( \sqrt{64 a^3 b^6} \), the square root implies a \( \frac{1}{2} \) exponent, hence you can rewrite it as \( (64 a^3 b^6)^{\frac{1}{2}} \). This form is easier to manipulate algebraically.

Power of a Product Rule

This rule is extremely useful when dealing with expressions inside parentheses raised to a power. The power of a product rule says that when you raise a product to an exponent, it can be broken down into each individual component raised to that power. In algebraic terms, this is: \( (xy)^n = x^n \cdot y^n \).

In our case, applying this rule to the expression \( (64 a^3 b^6)^{\frac{1}{2}} \), we split it into \( 64^{\frac{1}{2}} \cdot (a^3)^{\frac{1}{2}} \cdot (b^6)^{\frac{1}{2}} \). This simplifies each part separately, and makes complicated expressions more manageable.

• Each base gets the exponent individually.
• Great for simplifying expressions involving multiple variables and constants.

This rule is a foundational tool for simplifying expressions, especially when dealing with various roots and powers.

Simplifying Expressions

Simplifying expressions involves breaking them down into the simplest form possible. Once you convert the expression into a product of powers, like \( 64^{\frac{1}{2}} \cdot (a^3)^{\frac{1}{2}} \cdot (b^6)^{\frac{1}{2}} \), each component can be simplified:

• \( 64^{\frac{1}{2}} = 8 \) since \( 8^2 = 64 \)
• \((a^3)^{\frac{1}{2}} = a^{\frac{3}{2}} \)
• \((b^6)^{\frac{1}{2}} = b^3 \)

After simplifying each part, combine them back into one statement: \( 8 \cdot a^{\frac{3}{2}} \cdot b^3 \). This is your simplified expression and distinctly shows each variable raised to a power without any roots involved. This kind of simplification is crucial because it helps make equations easier to solve, compare, or even just analyze further in mathematics.