Problem 51 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 47-52: Rewrite the expression using rational exponents. $$ \sqrt{x^{2}+y^{2}} $$

Step-by-Step Solution

Answer

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

Step 1: Identify the Expression Type

Notice that the given expression is a square root. It can be represented as a power of one-half. We need to rewrite the expression using rational exponents.

Step 2: Rewrite the Square Root as a Rational Exponent

The expression \(\sqrt{x^{2}+y^{2}}\) can be rewritten as \((x^{2}+y^{2})^{1/2}\). In this form, the square root is expressed using a rational exponent of \(1/2\).

Key Concepts

Square Roots

Square roots are fundamental concepts in mathematics. They represent a value that, when multiplied by itself, will give you the original number. In the expression \(\sqrt{x^{2}+y^{2}}\), you are trying to find a number, which when squared, equals \(x^{2}+y^{2}\).
To work with square roots in algebra efficiently, it's useful to understand their relationship with exponents.
Here's how you can think about square roots:

• The square root of a number \(a\) can be symbolized as \(\sqrt{a}\).
• This is equivalent to raising \(a\) to the power of \(1/2\), i.e., \(a^{1/2}\).

This exponent form is particularly helpful for manipulating expressions algebraically. It allows for easier combination and simplification in equations and makes it easier to apply algebraic rules.

Exponentiation

Exponentiation involves raising numbers to a power. It is a way to express repeated multiplication. For example, \(x^3\) means \(x \cdot x \cdot x\).
In the context of rational exponents, like \((x^{2}+y^{2})^{1/2}\), you are essentially multiplying the base, \(x^{2}+y^{2}\), by itself half a time. This means finding a kind of middle point between multiplication and division.
Key things to remember about exponentiation with rational exponents:

• When you see a power like \(a^{m/n}\), it is equivalent to \(n\)-th root of \(a\) raised to the power \(m\). In simpler terms, you take the root, then apply the exponent.
• Rational exponents follow the same laws as integer exponents: product of powers, power of a power, and quotient of powers.
• This flexibility in form makes algebraic manipulation more versatile, providing a unified method for handling roots and powers.

By understanding these concepts, you can simplify complex expressions and solve equations more effectively.

Algebraic Expressions

An algebraic expression involves numbers, variables, and operations. Consider expressions like \(x^2 + y^2\), which combine variables \(x\) and \(y\) using the mathematical operations of addition and multiplication.
Learning to manipulate algebraic expressions is crucial, as it forms the foundation of algebra. Here are some fundamental aspects to consider:

• Simplification: Reduce expressions to their simplest form, where possible, by combining like terms or applying mathematical operations.
• Equivalence: Recognize when two expressions represent the same quantity, even if they look different.
• Substitution: Replace variables with numerical values to evaluate the expression for different cases.

Converting roots to rational exponents, as we did with \(\sqrt{x^{2}+y^{2}}\) to \((x^{2}+y^{2})^{1/2}\), is part of working with algebraic expressions. This conversion serves various purposes, such as simplifying calculations and facilitating further algebraic operations. Knowing how to express expressions in different forms allows for greater flexibility in problem-solving.