Problem 51 - Precalculus: Functions and Graphs

Question

Rewrite the expression using rational exponents. $$\sqrt{x^{2}+y^{2}}$$

Step-by-Step Solution

Answer

\((x^2 + y^2)^{1/2}\)

Step 1: Understand the Expression

The given expression is \( \sqrt{x^2 + y^2} \), which represents the square root of the sum of \( x^2 \) and \( y^2 \). The goal is to express this using rational exponents.

Step 2: Recall the Rule for Square Roots

Square roots can be rewritten using rational exponents. By definition, the square root of any expression \( a \) can be expressed as \( a^{1/2} \). This is a key property that we will apply to the given expression.

Step 3: Apply the Rule to the Given Expression

Using the property from the previous step, replace the square root in the expression with a rational exponent. Therefore, \( \sqrt{x^2 + y^2} \) can be rewritten as \((x^2 + y^2)^{1/2}\).

Key Concepts

Square Roots

Square roots are a fundamental concept in mathematics and are used to find a number which, when multiplied by itself, gives the original number. The square root of a number is represented by the radical symbol (√). An example is √4, which equals 2, since 2 multiplied by itself equals 4.

In mathematical expressions, the square root can be seen as undoing the operation of squaring a number. For instance, if you square 3, you get 9. The square root of 9 brings you back to your original number, which is 3. This idea of 'undoing' is central to understanding operations involving roots.

When working with square roots in equations, it’s important to simplify them when possible. This involves recognizing perfect squares and reducing the square root to its simplest form, which helps make more complex calculations manageable.

Expressions

Expressions in mathematics are combinations of numbers, variables, and mathematical operators (like addition and subtraction) that together specify a calculation to be performed. For instance, \(x^2 + y^2\) in the original exercise is an expression.

Expressions can be simple, like \(3x + 4\), or very complex, involving multiple operations. They can often be manipulated using algebraic rules to make them easier to solve.

Working with expressions requires knowledge of arithmetic operations, understanding of how terms interact, and simplifying wherever possible. For instance, combining like terms and factoring can transform expressions into more workable forms. This forms the basis for solving equations and can help in transforming expressions such as converting a square root into its equivalent form using exponents.

Exponent Rules

Exponent rules, also known as the laws of exponents, are crucial in working with any operation involving powers or roots. These rules provide a systematic way to handle powers across different terms and expressions.

Some essential exponent rules include:

• Product of Powers: \(a^m \times a^n = a^{m+n}\)
• Power of a Power: \((a^m)^n = a^{m \times n}\)
• Power of a Product: \((ab)^m = a^m \times b^m\)
• Zero Exponent Rule: \(a^0 = 1\), where \(a\) is not zero

Using these rules can help in simplifying expressions and are particularly useful when rewriting expressions in different forms, such as transforming square roots into rational exponents.

In our case, converting \(\sqrt{x^2 + y^2}\) to \((x^2 + y^2)^{1/2}\) utilizes the rational exponent idea, treating the square root as an exponent of 1/2, which allows for flexibility in further algebraic manipulation.

Mathematical Notation

Mathematical notation is a symbolic language used to convey mathematical concepts in a consistent and concise way. It includes symbols for numbers, variables, operations, and more complex concepts like functions.

This notation helps in expressing complex ideas efficiently. For example, symbols like \(\sqrt{}\) for square roots, or \(^{}\) for powers, can replace lengthy verbal descriptions, making it easier to work through problems.

Well-understood notation is essential for communication in mathematics, allowing mathematicians anywhere to understand calculations without ambiguity.

• Rational exponents, such as writing \(a^{1/2}\) for \(\sqrt{a}\), combine simplicity with functionality, demonstrating how simple notations can convey deeper concepts.
• The use of notation helps with standardization, ensuring that everyone uses the same language and symbols for the same concepts.

Understanding and properly using mathematical notation allows students to follow mathematical arguments and procedures accurately, providing the tools needed to succeed in any math-related field.