Problem 51 - A Graphical Approach to College Algebra

Question

Incorporate many concepts from Chapter 3 with the method of solving equations involving roots. Work them in order. Consider the equation $$\sqrt[3]{4 x-4}=\sqrt{x+1}$$ Rewrite the equation, using rational exponents.

Step-by-Step Solution

Answer

Rewrite: \((4x - 4)^{1/3} = (x + 1)^{1/2}\).

Step 1: Understand the Problem

The given equation is \(\sqrt[3]{4x - 4} = \sqrt{x + 1}\). Our goal is to rewrite this equation using rational exponents.

Step 2: Convert Radical to Rational Exponents

Recall that \(\sqrt[n]{a} = a^{1/n}\). Apply this to each side of the equation:- \(\sqrt[3]{4x - 4}\) becomes \((4x - 4)^{1/3}\).- \(\sqrt{x + 1}\) becomes \((x + 1)^{1/2}\).This gives us the equation \((4x - 4)^{1/3} = (x + 1)^{1/2}\).

Step 3: Review Result

The rewritten equation using rational exponents is \((4x - 4)^{1/3} = (x + 1)^{1/2}\). This is the desired format we need for further solving.

Key Concepts

Solving Equations

Solving equations efficiently is a crucial skill in algebra. It means finding the value of the variable that makes the equation true. To solve the equation \(\sqrt[3]{4x - 4} = \sqrt{x + 1}\), we rewrite it using rational exponents first. This format allows for better algebraic manipulation.Here’s a step-by-step for solving an equation:

• Identify the type of equation: Look for operations and what’s being solved for.
• Rewrite using simpler terms: Convert roots to exponents or vice versa.


• Isolate the variable: Perform operations to get the variable alone on one side.

In this context, we aim to simplify both sides so algebraic methods like squaring or taking powers can balance or cancel out terms.

Radicals

Radicals or roots involve finding a number which, when multiplied a certain number of times by itself, equals a given value. In our equation, we have two different radicals: the cube root \(\sqrt[3]{4x - 4}\) and the square root \(\sqrt{x + 1}\).To work with radicals, keep these points in mind:

• The cube root \(\sqrt[3]{a}\) implies a number \(b\) such that \(b^3 = a\).
• The square root \(\sqrt{b}\) means \(c\) such that \(c^2 = b\).

Converting to Rational Exponents: Recognizing radicals as fractional powers helps simplify and compare such expressions, as seen in rewriting \(\sqrt[3]{4x - 4}\) as \((4x - 4)^{1/3}\) and \(\sqrt{x + 1}\) as \((x + 1)^{1/2}\). This conversion facilitates same-base comparisons or operations by leveling the 'playing field' between differing radical operations.

Exponential Form

Exponential form is the representation of expressions using powers or exponents. This form is powerful in algebra as it makes sophisticated equations simpler to manipulate.In our example:

• We write the cube root \(\sqrt[3]{4x - 4}\) as \((4x - 4)^{1/3}\).
• The square root \(\sqrt{x + 1}\) becomes \((x + 1)^{1/2}\).

Why use exponential form? - This format helps in standardizing different types of equations and operations under the same rule sets.- Any operation done on one side can be easily mirrored on the other, especially when isolating variables.- It also allows clear application of rules like the power of a power rule, \((a^m)^n = a^{m \cdot n}\), necessary for solving equations.

Algebraic Manipulation

Algebraic manipulation involves the strategic use of algebraic rules to simplify equations or expressions. This skill is essential for shaping an equation into a form that makes finding solutions possible.In our equation \((4x - 4)^{1/3} = (x + 1)^{1/2}\), here’s how algebraic manipulation can proceed:

• Eliminate fractional exponents: Raise each side to a power that clears these fractions. For example, cube both sides to eliminate the fraction \(1/3\) or square both sides to eliminate \(1/2\), depending on what allows further simplification.
• Balance the equation: Always perform the same operation on both sides. This step maintains equality.


• Simplify: Combine like terms or simplify exponents to reveal or isolate the variable.

These steps systematically reduce complex equations to simple forms where the variable is easily identified.