Problem 55 - Algebra for College Students

Question

Write each of the following using positive rational exponents. For example, \(\sqrt{a b}=(a b)^{\frac{1}{2}}=a^{\frac{1}{2}} b^{\frac{1}{2}}\). \(5 x \sqrt{y}\)

Step-by-Step Solution

Answer

\(5x^1y^{\frac{1}{2}}\)

Step 1: Express the Square Root as an Exponent

The square root of a number is the same as raising that number to the power of \(\frac{1}{2}\). For the expression \(5x\sqrt{y}\), we can rewrite \(\sqrt{y}\) as \(y^{\frac{1}{2}}\). This changes the expression to \(5x \cdot y^{\frac{1}{2}}\).

Step 2: Apply the Property of Exponents

According to the properties of exponents, a multiplication of terms can individually have exponents applied to each. Thus, further simplifying, we have \(5 \cdot x^1 \cdot y^{\frac{1}{2}}\). Note that \(x^1\) represents that \(x\) is already in an exponent form even though it's commonly omitted.

Key Concepts

Properties of Exponents

Understanding the properties of exponents can make working with complex expressions much simpler. Exponents allow us to express repeated multiplication efficiently. Some rules are fundamental to working with exponents effectively:

• Product of Powers: When multiplying two exponents with the same base, you add their powers. For example, \( a^m \times a^n = a^{m+n} \).
• Power of a Power: When raising an exponent to another exponent, you multiply the exponents. For example, \( (a^m)^n = a^{m \times n} \).
• Power of a Product: When raising a product to an exponent, each factor is raised to the exponent individually. For example, \( (ab)^m = a^m \times b^m \).

When simplifying expressions with exponents, these rules help us transform and simplify terms efficiently. In our problem, we used these properties to rewrite and simplify the expression involving a square root and variables.

Simplifying Expressions

Simplifying expressions is about making them easier to work with, without changing their value. It often involves combining like terms and applying algebraic rules like those for exponents. In the expression \(5x\sqrt{y}\), simplifying involves a few steps:

• Convert Square Roots: Recognize that a square root can be expressed as an exponent. \(\sqrt{y}\) becomes \(y^{\frac{1}{2}}\).
• Separate Terms with Exponents: Apply the power of a product property to show each factor with its own exponent: \(5 \cdot x \cdot y^{\frac{1}{2}}\).
• Combine and Organize: Recognize that \(x^1\) signifies \(x\) has an implicit exponent, helping in later calculations.

By doing these steps, the expression becomes just a collection of multiplications with clear exponents, showing relationships between the variables and constants.

Square Roots as Exponents

Square roots are fundamental in algebra, but they can often seem tricky to combine with other numbers. One helpful concept is transforming them into exponents for easier manipulation. In mathematical terms:

• Exponent Representation: The square root of any number \(a\) is the same as raising \(a\) to the power of \(\frac{1}{2}\). Thus, \(\sqrt{a} = a^{\frac{1}{2}}\).
• Simplifying Using Properties: With roots as exponents, you can apply the same properties as any other exponent. This means easier multiplication and division.

This conversion shines in expressions like \(5x\sqrt{y}\), where \(\sqrt{y}\) becomes \(y^{\frac{1}{2}}\), aligning it with the other exponents in the equation. Overall, expressing square roots as exponents streamlines problem-solving, giving a uniform approach to handling terms.