Problem 60 - Elementary and Intermediate Algebra

Question

Change each radical to an exponential expression. See Example 5. $$ \sqrt{x^{2}+y^{2}} $$

Step-by-Step Solution

Answer

\( \sqrt{x^2 + y^2} \) is \( (x^2 + y^2)^{\frac{1}{2}} \).

Step 1: Analyze the problem

Identify the type of problem and the appropriate mathematical technique to apply.

Step 2: Apply the technique and solve

\( \sqrt{x^2 + y^2} \) is \( (x^2 + y^2)^{\frac{1}{2}} \)..

Step 3: Verify the result

Check the answer by substitution or alternative methods to confirm correctness.

Key Concepts

Radical Expressions

In mathematics, radical expressions are expressions that involve roots, such as square roots or cube roots. A radical expression usually includes a radical sign, denoted as \( \sqrt{} \), to indicate the root of a quantity. For example, \( \sqrt{4} \) represents the square root of 4.

Understanding radical expressions is important because they serve as a foundation for various mathematical operations, including simplifying expressions and solving equations. When working with radicals:

• The number inside the radical sign is known as the "radicand." In \( \sqrt{x^2 + y^2} \), the radicand is \( x^2 + y^2 \).
• The root indicated by the radical can vary. The square root \( \sqrt{} \) implies a 2nd root, but radicals can indicate other roots like cube roots (3rd root), denoted as \( \sqrt[3]{} \).
• Radicals can be rewritten as exponents, which is especially useful when solving algebraic expressions or equations, setting the stage for exponent rules to be applied.

Renaming radicals as exponentials helps in performing advanced calculations.

Fractional Exponents

Fractional exponents are another way to express roots using powers. Instead of using a radical sign, roots can be expressed by the numerator and the denominator in the exponent. The denominator of a fractional exponent represents the root, while the numerator represents the power the radicand is raised to inside the radical.

For instance, converting a square root to a fractional exponent works like this:

• \( \sqrt{a} \) is equivalent to \( a^{\frac{1}{2}} \).
• The number 2 in the denominator indicates the square root.
• Fractional exponents make it easier to apply algebraic rules because you can use the same operations for these exponents as you would with integers.

Thus, \( \sqrt{x^2 + y^2} \) becomes \( (x^2 + y^2)^{\frac{1}{2}} \). This transformation simplifies the manipulation of the expression, enabling the use of exponential rules for simplifying, differentiating, or integrating the expression if needed.

Square Roots

The concept of square roots is fundamental in understanding how linear and area measurements relate. A square root is a special instance of a radical where the root is 2, meaning you seek a number which, when multiplied by itself, yields the radicand. For instance, the square root of 9 is 3 because \( 3 \times 3 = 9 \).

When dealing with expressions like \( \sqrt{x^2 + y^2} \):

• The expression encapsulates both variables \( x \) and \( y \) within a square root, highlighting the importance of finding dimensions in geometric contexts, such as the hypotenuse in the Pythagorean theorem.
• Square roots simplify complex mathematical problems by offering a way to switch from lengthy multiplication to simpler exponential operations.
• In physics and engineering, square roots frequently appear in formulas involving measurements like velocity and acceleration, exemplifying their practical utility.

By converting square roots to exponential form, one can streamline calculations and apply consistent rules for manipulating expressions effectively.