Problem 22 - A Graphical Approach to Precalculus with Limits

Question

Use positive rational exponents to rewrite each expression. Assume variables represent positive numbers. $$\sqrt{x+1}$$

Step-by-Step Solution

Answer

\( \sqrt{x+1} = (x+1)^{1/2} \)

Step 1: Understanding the Square Root

The square root of a number or expression can be written as that number raised to the power of 1/2. For example, the square root of "a" is written as "a" to the power of 1/2. Thus, the expression \( \sqrt{x+1} \) can be rewritten using exponents.

Step 2: Convert the Square Root to Exponential Form

Now that we understand the square root can be rewritten using exponents, we convert the expression \( \sqrt{x+1} \) into exponential form. We use the rule \( \sqrt{a} = a^{1/2} \) to rewrite the expression as \( (x+1)^{1/2} \).

Key Concepts

Square Root

A square root is a number that, when multiplied by itself, produces the original number. For instance, the square root of 16 is 4 because 4 times 4 equals 16. In mathematical expressions, the square root is often denoted by a radical symbol \( \sqrt{} \). However, it can also be expressed using exponents. This is particularly useful when simplifying equations or expressions.
When you see \( \sqrt{a} \), it indicates the square root of \( a \). The equivalent expression using exponents is \( a^{1/2} \). This representation is crucial when dealing with more complex mathematical problems, such as those involving rational exponents or polynomial expressions.
Understanding this concept allows you to manipulate and simplify a variety of algebraic expressions more efficiently.

Exponentiation

Exponentiation is a fundamental mathematical operation akin to repeated multiplication. For instance, \( a^n \) means multiplying \( a \) by itself \( n \) times. It's a shorthand way to express the repeated multiplication of a number by itself.
There are special scenarios, such as multiplying by fractions, which are referred to as rational exponents. Rational exponents involve exponents that are fractions. The top number (numerator) represents the power, and the bottom number (denominator) represents the root. Therefore, \( a^{1/2} \) simplifies to the square root of \( a \). Here the numerator is 1, which means we don't "raise" \( a \) to any power beyond itself, and the denominator is 2, indicating the square root.

• A positive exponent means multiplying the base \( a \) by itself.
• Fractional exponents signify roots; e.g., \( a^{m/n} \) stands for the \( n \)th root of \( a \) raised to the power \( m \).
• Understanding how to manipulate exponentiation is invaluable for solving complex equations involving exponents and roots.

Positive Numbers

Positive numbers are numbers greater than zero. They are located to the right of zero on the number line and include fractions, whole numbers, and decimals that are more than zero. In the realm of rational exponents, positive numbers ensure that we avoid complications like undefined results when dealing with roots.
Why is positivity important here? When variables represent positive numbers, especially in inequalities or when finding roots, it ensures that expressions like \( \sqrt{x+1} \) produce real, defined results. With each number being positive, we follow straightforward arithmetic rules without encountering undefined expressions or negative roots.

• Positive numbers guarantee well-defined results for square roots.
• In rational exponent notation, it helps in maintaining the integrity of operations and results.
• Always remember, variables often assumed in academic problems are positive unless stated otherwise, simplifying many mathematical processes.

By keeping numbers positive, the operations remain coherent and lead to meaningful outcomes when working with roots and exponents.