Problem 89 - A Graphical Approach to Precalculus with Limits

Question

Incorporate many concepts from earlier work 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

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

Step 1: Understand the problem

We are given the equation \( \sqrt[3]{4x - 4} = \sqrt{x + 1} \). Our task is to rewrite this equation using rational exponents instead of radical symbols.

Step 2: Convert cube root to rational exponent

The cube root \( \sqrt[3]{4x - 4} \) can be written using rational exponents. The rule is that \( \sqrt[3]{a} = a^{1/3} \). Therefore, \( \sqrt[3]{4x - 4} = (4x - 4)^{1/3} \).

Step 3: Convert square root to rational exponent

Similarly, the square root \( \sqrt{x + 1} \) can be expressed with rational exponents using the rule \( \sqrt{a} = a^{1/2} \). Hence, \( \sqrt{x + 1} = (x + 1)^{1/2} \).

Step 4: Rewrite the original equation

Now replace the radical expressions in the original equation with their rational exponent forms. The equation \( \sqrt[3]{4x - 4} = \sqrt{x + 1} \) becomes \( (4x - 4)^{1/3} = (x + 1)^{1/2} \).

Key Concepts

Radicals

Radicals are expressions that involve roots, such as square roots, cube roots, and so on. They are a way to represent numbers or expressions that, when raised to a certain power, produce the original number. For example, the square root of a number is a value that, when multiplied by itself, gives the original number. Radicals are often represented with the radical symbol \( \sqrt{} \) followed by the expression under the root.

There are different types of radicals based on the degree of the root:

• Square Roots: The most common type, represented as \( \sqrt{a} \), which means \( a^{1/2} \).
• Cube Roots: Represented as \( \sqrt[3]{a} \), meaning \( a^{1/3} \).
• Fourth Roots, Fifth Roots, etc.: Continue in a similar fashion, such as \( \sqrt[4]{a} \), meaning \( a^{1/4} \).

To simplify mathematical operations, radicals can be expressed using rational exponents, where the root turns into a fractional exponent. This transformation makes it easier to perform operations like multiplication and division on radicals.

Equations

Equations are mathematical statements that assert the equality of two expressions. They are the foundation of algebra and are used to find unknown values. The equation given in the exercise, \( \sqrt[3]{4x - 4} = \sqrt{x + 1} \), connects two radical expressions. Solving equations often involves manipulating both sides to isolate the unknown variable, in this case, \( x \).

In order to solve equations involving radicals, one useful method is to convert radicals into expressions with rational exponents. This makes it easier to apply algebraic rules such as raising both sides to a power to eliminate the irrational exponents.

Here is a simple strategy to tackle such equations:

• Step 1: Convert all radical expressions to rational exponents.
• Step 2: Raise both sides of the equation to eliminate fractions in exponents.
• Step 3: Simplify and solve for the variable.

This process is often referred to as rationalizing the equation and is crucial for finding the solution to complex equations involving radicals.

Cube Roots

Cube roots are a special type of radical. They are used when you need to find a number that, when raised to the third power, equals the original number. The cube root is represented by \( \sqrt[3]{} \). For instance, the cube root of 8 is 2 because \( 2^3 = 8 \).

Cube roots have a unique property in that every real number has one real cube root, which can be either positive, negative, or zero. This is unlike square roots, where a negative number does not have a real square root.

Expressing cube roots as rational exponents is straightforward. As per the rule \( \sqrt[3]{a} = a^{1/3} \), you can easily transform a cube root expression into an exponent form. This transformation simplifies the process of solving and engaging with algebraic expressions.

Understanding cube roots and their transformations into rational exponents is essential for dealing with complex algebraic equations, especially those that need to be simplified or solved.

Square Roots

Square roots are one of the most elementary forms of radicals. They signify a value that, when squared, will result in the original number. The square root operation is denoted by \( \sqrt{} \), and an example is \( \sqrt{9} = 3 \), since \( 3^2 = 9 \).

Square roots are commonly found in algebra and have properties that make them especially useful in solving quadratic equations. They can, however, complicate an equation, which is why converting square roots to rational exponents, such as \( a^{1/2} \), is a useful strategy.

Here are key points about square roots:

• Every positive number has two square roots: a positive and a negative one.
• The primary focus is often on the non-negative root, known as the principal square root.
• Square roots are not defined for negative numbers within the real number system, but in algebra, they extend to complex numbers.

Converting square roots to exponent form simplifies operations such as multiplication and division. This conversion is especially helpful in solving equations that involve both square roots and other algebraic operations.