Problem 47 - Algebra for College Students

Question

Write each of the following using positive rational exponents. For example, \(\sqrt{a b}=(a b)^{\frac{1}{2}}=a^{\frac{1}{2}} b^{\frac{1}{2}}\). \(3 \sqrt{y}\)

Step-by-Step Solution

Answer

\(3 \sqrt{y} = 3 y^{\frac{1}{2}}\)

Step 1: Identify the radical expression

The expression given is \(3 \sqrt{y}\), which involves a square root.

Step 2: Write the square root as an exponent

The square root \(\sqrt{y}\) can be expressed using the rational exponent notation as \(y^{\frac{1}{2}}\).

Step 3: Combine the rational exponent with the coefficient

Since \(3 \sqrt{y}\) is equivalent to \(3 \times y^{\frac{1}{2}}\), we can write the expression as \(3 y^{\frac{1}{2}}\).

Key Concepts

Radical Expressions

Radical expressions represent mathematical expressions that include a root symbol. These expressions typically involve operations that require finding square roots, cube roots or higher order roots of a number.
Often, they look something like \( \sqrt{x} \), where the symbol \( \sqrt{} \) denotes a square root, or \( \sqrt[3]{x} \) for a cube root.

• The number or expression under the root is called the radicand. For example, in \( \sqrt{y} \), \( y \) is the radicand.
• The index of the root is essential because it tells you which root you’re dealing with. A missing index suggests a square root.

Using rational exponents instead of radical notation can make more complex expressions easier to manage. Rational exponents allow us to express roots without using the radical symbol, providing a more uniform way to handle powers and roots in further mathematical operations.

Square Root

The square root symbolizes a value that when multiplied by itself results in the original number. For example, the square root of 9 is 3 because \(3 \times 3 = 9\).
It is written using the radical symbol \(\sqrt{}\), but it can also be expressed using exponents: \(\sqrt{y} = y^{\frac{1}{2}}\).
Understanding square roots is fundamental because they are one of the most common types of roots you’ll encounter.

• Square roots are only defined for non-negative numbers in the set of real numbers, as negative numbers under the square root give imaginary numbers.
• Converting a square root to a rational exponent makes it easier to apply algebraic rules or integrate it into broader algebraic expressions.

Square roots can greatly simplify the solving of equations, especially those involving quadratic terms or radical simplifications.

Exponents

Exponents allow us to express repeated multiplication of a base number. The notation \(y^n\) indicates that the base \(y\) is multiplied by itself \(n\) times.
They can be whole numbers, fractions, or negatives, each serving different purposes:

• Whole number exponents represent straightforward multiplication: \(y^3 = y \times y \times y\).
• Fractional exponents, like \(y^{\frac{1}{2}}\), represent roots: it's shorthand for the square root of \(y\).
• Negative exponents indicate division: \(y^{-n} = \frac{1}{y^n}\).

Exponents play a crucial role in simplifying expressions and solving equations. With a unified exponent notation for both powers and roots, it is easier to perform operations such as multiplication, division, and distribution across terms.
This makes rational exponents a powerful tool in algebra and higher-level mathematics.