Problem 53 - Precalculus: Functions and Graphs

Question

Rewrite the expression using a radical. (a) \(4 x^{3 / 2}\) (b) \((4 x)^{3 / 2}\)

Step-by-Step Solution

Answer

(a) \(4x\sqrt{x}\), (b) \(8x\sqrt{x}\)

Step 1: Understand the Fractional Exponent

The fractional exponent \(\frac{3}{2}\) indicates a radical and a power. The denominator (2) implies a square root, while the numerator (3) indicates that the base is raised to the third power.

Step 2: Rewrite (a) as Radical Expression

For part (a), \(4 x^{3/2}\), rewrite the expression by first finding the radical for \(x\). \(x^{3/2}\) can be rewritten as \((x^{1/2})^3\) or \(\sqrt{x}^3\). Thus, the expression \(4 x^{3/2}\) becomes \(4 \sqrt{x^3}\) which simplifies further to \(4(\sqrt{x})^3\).

Step 3: Simplify Expression (a)

Using the laws of exponents and radicals, \(4\sqrt{x^3}\) simplifies to \(4x\sqrt{x}\), which is the simplest radical form for part (a).

Step 4: Rewrite (b) as Radical Expression

For part (b), \((4x)^{3/2}\), rewrite it as \(((4x)^{1/2})^3\) or \((\sqrt{4x})^3\). This represents the expression where the entire base \(4x\) is under the square root and then raised to the third power.

Step 5: Simplify Expression (b)

\(\sqrt{4x} = \sqrt{4} \times \sqrt{x} = 2\sqrt{x}\). Thus, \((\sqrt{4x})^3 = (2\sqrt{x})^3 = 2^3 \cdot (\sqrt{x})^3 = 8x\sqrt{x}\), which is the simplest radical form for part (b).

Key Concepts

Fractional Exponents

Fractional exponents are a way to represent roots and powers using fractions. In these expressions, the numerator represents the power to which the base is raised, while the denominator indicates the type of root, such as a square root or cube root. For example, in the expression \(x^{3/2}\), the denominator 2 indicates a square root, and the numerator 3 signifies that the base \(x\) is then raised to the power of 3.

Key points to remember when working with fractional exponents:

• A fractional exponent means a combination of a root and a power.
• The denominator specifies the root (e.g., 2 for square root, 3 for cube root).
• The numerator specifies how many times the base is multiplied by itself.

By understanding these components, we can rewrite fractional exponents into radical expressions, making it easier to manipulate and simplify complex algebraic expressions.

Simplifying Radicals

Simplifying radicals involves rewriting a radical expression into its simplest form. When simplifying radicals such as \(\sqrt{x^3}\), we aim to break down the expression into its simplest terms, often by finding perfect squares within the radicand.

Consider the radical expression \(\sqrt{x^3}\). This can be rewritten by separating the square roots:

• Break down \(x^3\) into \(x^2 \times x\).
• This allows us to write it as \(\sqrt{x^2 \times x} = \sqrt{x^2} \times \sqrt{x} = x\sqrt{x}\).

In this way, we can consistently simplify radicals by identifying perfect square factors and reducing them outside the radical sign for a cleaner expression. Simplifying radicals is integral in making expressions easier to work with, especially when dealing with calculations or further algebraic operations.

Laws of Exponents

The laws of exponents are fundamental rules that govern how exponential expressions can be manipulated. Knowing these laws helps in simplifying expressions and solving equations involving exponents or powers.

Some key laws of exponents include:

• Product of Powers: \(a^m \cdot a^n = a^{m+n}\)
• Power of a Power: \((a^m)^n = a^{m \cdot n}\)
• Power of a Product: \((ab)^n = a^n \cdot b^n\)
• Quotient of Powers: \(\frac{a^m}{a^n} = a^{m-n}\)

In the given problem, the power of a product law is particularly useful. For instance, when simplifying \((\sqrt{4x})^3\), we use the property that each part can be powered separately: first find \(\sqrt{4x} = 2\sqrt{x}\), then raise each component to the third power, yielding \(8x\sqrt{x}\). These laws are powerful tools that make managing and simplifying expressions clear and systematic.