Problem 19 - College Algebra with Modeling and Visualization

Question

Use positive exponents to rewrite. $$ \sqrt{2 x} $$

Step-by-Step Solution

Answer

\( \sqrt{2x} = 2^{\frac{1}{2}} \times x^{\frac{1}{2}} \)

Step 1: Understand the Square Root Notation

The square root notation \( \sqrt{x} \) is equivalent to raising the number \( x \) to the power of \( \frac{1}{2} \). This transformation can be very useful when working with exponents.

Step 2: Apply the Exponent Rule

Using the understanding from Step 1, rewrite \( \sqrt{2x} \) as \( (2x)^{\frac{1}{2}} \). The expression now uses an exponent instead of a square root.

Step 3: Separate the Factors

The property of exponents allows us to apply the exponent to both components of the product inside the parentheses. Therefore, \( (2x)^{\frac{1}{2}} \) can be rewritten as \( 2^{\frac{1}{2}} \times x^{\frac{1}{2}} \).

Key Concepts

Exponent Rule

Exponents are powerful tools in algebra that allow us to express repeated multiplication in a more compact form. The exponent tells us how many times to multiply the base by itself. For example, in the expression \( a^n \), \( a \) is the base and \( n \) is the exponent.
When working with exponents, some important rules apply:

• Product Rule: If you multiply two powers with the same base, you add the exponents: \( a^m \times a^n = a^{m+n} \).
• Quotient Rule: If you divide two powers with the same base, you subtract the exponents: \( \frac{a^m}{a^n} = a^{m-n} \).
• Power of a Power Rule: If you raise a power to another power, you multiply the exponents: \( (a^m)^n = a^{m \times n} \).

These rules help in simplifying expressions and solving equations that involve exponents. Even when dealing with fractional exponents, such as \( (2x)^{\frac{1}{2}} \), the rules remain true. Understanding these will make manipulating exponents much easier, and they are especially useful in transitioning between different forms like from a square root notation to an exponent notation.

Square Root Notation

The square root notation is a fundamental concept in algebra, denoted by the radical symbol \( \sqrt{} \). When you see \( \sqrt{x} \), it indicates the principal square root of \( x \), which is the positive number that, when multiplied by itself, equals \( x \).
An essential piece of knowledge is that a square root can also be expressed using fractional exponents. Specifically, \( \sqrt{x} \) is equivalent to \( x^{\frac{1}{2}} \). This notion is incredibly useful because it allows us to apply exponent rules to simplify expressions and solve equations.
For instance, converting \( \sqrt{2x} \) into \( (2x)^{\frac{1}{2}} \) enables us to use all the powerful properties of exponents to further manipulate the expression. Recognizing this conversion is a key step in solving many algebraic problems efficiently.

Positive Exponents

Positive exponents are straightforward as they indicate the number of times a base is to be multiplied by itself. A positive exponent like \( x^2 \) means that \( x \) is multiplied by itself, giving you \( x \times x \).
In algebra, our goal is often to express terms using positive exponents because they are the simplest and most intuitive form.
By transforming expressions such as \( \sqrt{2x} \) into \( (2x)^{\frac{1}{2}} \), we switch from a root notation to an exponent notation using positive exponents, making them simpler to handle with the exponent rules. The expression \( x^{\frac{1}{2}} \), though a fractional exponent, is still a positive exponent because it's applied repeatedly in a positive factor approach. Careful manipulation of these expressions ensures clarity and precision in algebraic operations.