Problem 54 - 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) \(4 + \sqrt{x^3}\); (b) \(\sqrt{(4 + x)^3}\)

Step 1: Identify the Expression Type

Determine if each part of the expression contains a term that can be rewritten using a radical. For both expressions (a) and (b), note that the exponent \(\frac{3}{2}\) indicates a radical transformation is possible.

Step 2: Address Expression (a)

For the expression \(4 + x^{3/2}\), identify the \(x^{3/2}\) term which can be rewritten using a radical. The power \(x^{3/2}\) can be broken down into the square root of \(x^3\), which is \(\sqrt{x^3}\) or \((\sqrt{x})^3\). Thus, the expression becomes \(4 + \sqrt{x^3}\).

Step 3: Address Expression (b)

For the expression \((4 + x)^{3/2}\), understand that the entire term \((4 + x)\) is raised to the power of \(\frac{3}{2}\). This can be expressed as \((4 + x)\) raised to the power \(3\) over \(2\), which translates to \(\sqrt{(4 + x)^3}\). This results from taking the square root of the entire expression raised to the power of 3.

Key Concepts

Rational Exponents

Rational exponents may initially seem complex, but they are simply another way to represent radicals. A rational exponent is an exponent that is a fraction. The most common example is \[x^{\frac{1}{n}}\] which translates to the nth root of \(x\). For instance, an exponent of \(\frac{1}{2}\) corresponds to a square root, because it represents the 2nd root of a number. If you see an expression like \(x^{\frac{3}{2}}\), it indicates taking the square root of \(x^3\).

• The numerator (3, in this case) works as a regular exponent.
• The denominator (2) tells you the root level.

This form allows for easier manipulation and can simplify many algebraic expressions, making calculations more efficient.

Square Roots

Square roots are one of the most common types of radical expressions. They undo squaring and represent a value that, when squared, equals the original number. The square root symbol \(\sqrt{}\) indicates this operation. For example, the square root of 9 is 3, because 3 multiplied by itself gives 9. Similarly, if you have \(x^{3/2}\), we can break it down to:\[\sqrt{x^{3}}\]This means you first find \(x^3\) and then take the square root of that result. Recognizing square roots in rational exponent form is key:

• They simplify complex expressions.
• They are integral in solving quadratic equations and other algebraic formulas.

Understanding how to convert between radicals and rational exponents can expand your ability to handle various mathematical challenges effectively.

Exponentiation

Exponentiation is the process of taking a base number and raising it to an exponent, which dictates how many times to multiply the base by itself. When dealing with expressions like \((4 + x)^{3/2}\), exponentiation becomes a bit more nuanced. Here, the fractional exponent indicates both raising \((4 + x)\) to the third power and then taking the square root of the result:- **Calculating the Power:** First, compute \((4 + x)^3\), multiplying the sum by itself three times.- **Applying the Root:** Next, take the square root of this entire expression, which is indicated by the exponent denominator, 2.This process underscores the flexibility of exponent laws, allowing transformations from radical to exponential form and vice versa, assisting in simplifying and solving various algebraic problems.

Algebraic Transformation

Algebraic transformations involve changing the form of an equation or expression to make it easier to work with or solve. This often includes switching between different forms, such as moving from rational exponents to radical notation, as demonstrated in the given problem:- **Expression (a):** Transform \(x^{3/2}\) into \(\sqrt{x^3}\) to emphasize the radical operation.- **Expression (b):** Convert \((4 + x)^{3/2}\) into \(\sqrt{(4 + x)^3}\), clarifying the operations needed.Such transformations help in solving or simplifying equations, revealing different perspectives on a problem. Using algebraic transformations, seemingly complex problems can become manageable, highlighting their efficiency in algebraic solutions. They allow for effective simplification techniques that make expressions more comprehensible and solutions more direct, creating clear pathways in mathematical problem-solving.