Problem 45 - Algebra for College Students

Question

For Problems \(45-58\), write each of the following using positive rational exponents. For example, \(\sqrt{a b}=(a b)^{\frac{1}{2}}=a^{\frac{1}{2}} b^{\frac{1}{2}}\) \(\sqrt{5 y}\)

Step-by-Step Solution

Answer

\( \sqrt{5y} = 5^{\frac{1}{2}} y^{\frac{1}{2}} \)

Step 1: Understanding the Problem

We need to write \( \sqrt{5y} \) using positive rational exponents. This involves rewriting the square root expression in terms of a fractional exponent.

Step 2: Rewriting the Square Root

The square root of an expression can be represented using a rational exponent where the exponent is \( \frac{1}{2} \). So, we can write \( \sqrt{5y} \) as \( (5y)^{\frac{1}{2}} \).

Step 3: Applying the Property of Exponents

According to the property of exponents \( (ab)^{m} = a^{m}b^{m} \), we can distribute the exponent \( \frac{1}{2} \) to both \( 5 \) and \( y \). Therefore, \((5y)^{\frac{1}{2}} = 5^{\frac{1}{2}} y^{\frac{1}{2}}\).

Key Concepts

Square Roots

Square roots are one of the most fundamental concepts in mathematics, providing the foundation for many more advanced topics. A square root of a number is essentially a value that, when multiplied by itself, gives back the original number. For example,

• The square root of 4 is 2 because 2 multiplied by 2 equals 4.
• Similarly, the square root of 9 is 3, since 3 times 3 equals 9.

In terms of notation, the square root of a number or expression can be expressed as a radical symbol (√). Square roots are closely related to exponents, particularly when expressed using rational exponents. When dealing with expressions under a root, the square root can be converted into an exponent of \( \frac{1}{2} \). Thus, \( \sqrt{n} = n^{\frac{1}{2}} \). This transformation forms the backbone of solving problems involving square roots using rational exponents. It makes it easier to apply algebraic rules such as simplifying or solving equations.

Properties of Exponents

Exponents form the basis of many mathematical rules and operations. Understanding the properties of exponents can simplify complex calculations. Here are some key properties:

• 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: \( (ab)^m = a^m b^m \)
• Negative Exponent: \( a^{-m} = \frac{1}{a^m} \)

Each of these rules helps in breaking down and simplifying expressions. For example, in our original exercise, by distributing the exponent of \( \frac{1}{2} \) over the product \( 5y \), we utilized the Power of a Product property to rewrite \( (5y)^{\frac{1}{2}} \) as \( 5^{\frac{1}{2}} y^{\frac{1}{2}} \). This property is particularly useful when dealing with polynomials and products under a radical.

Fractional Exponents

Fractional exponents are a concise way of representing roots and powers in mathematical expressions. The most general form of a fractional exponent is \( b^{\frac{m}{n}} \), where:

• \( b \) is the base,
• \( m \) is the power the base is raised to, and
• \( n \) is the root of the base.

Expressing a root with an exponent simplifies many calculations and allows the utilization of exponent properties to manipulate the expression. For example, in the expression \( (5y)^{\frac{1}{2}} \), the exponent \( \frac{1}{2} \) indicates that both the base 5 and the variable \( y \) are under a square root because \( \sqrt{a} \) is equivalent to \( a^{\frac{1}{2}} \). This method of using fractional exponents is not only cleaner in notation but also streamlines calculations when applying algebraic rules.