Problem 3 - Algebra 2 and Trigonometry

Question

In \(3-37,\) express each power as a rational number in simplest form. $$ 4^{\frac{1}{2}} $$

Step-by-Step Solution

Answer

The rational number is 2.

Step 1: Understanding the Problem

The exercise asks us to express the power \( 4^{\frac{1}{2}} \) as a rational number in its simplest form. A power in the form \( a^{\frac{1}{b}} \) is the same as a root, specifically \( \sqrt[b]{a} \).

Step 2: Recognize the Meaning of the Exponent

In \( 4^{\frac{1}{2}} \), the exponent \( \frac{1}{2} \) indicates that we need to find the square root of 4. This is because raising a number to the \( \frac{1}{2} \) power is the same as taking the square root of that number.

Step 3: Calculate the Square Root

The square root of 4 is calculated as \( \sqrt{4} \). We know that \( \sqrt{4} = 2 \), since \( 2 \times 2 = 4 \). Thus, \( 4^{\frac{1}{2}} = 2 \).

Step 4: Express the Result as a Rational Number

The result \( 2 \) is already a rational number, as it can be expressed as a fraction: \( \frac{2}{1} \), which is its simplest form.

Key Concepts

Understanding Exponents

Exponents are a shorthand way of expressing repeated multiplication. An exponent consists of a base number and a power, where the power tells us how many times to multiply the base number by itself. For example, in the expression \(a^n\), \(a\) is the base and \(n\) is the exponent. This means you multiply \(a\) by itself \(n\) times.
A special case of exponents is when we have a fractional exponent, such as \(4^{\frac{1}{2}}\). Here, the numerator (the top number in the fraction) acts like a regular exponent, while the denominator (the bottom number) indicates which root of the base we need. Thus, \(4^{\frac{1}{2}}\) becomes the square root of \(4\), because \(\frac{1}{2}\) indicates the second root. Understanding these fractional exponents is crucial for simplifying expressions.

Demystifying Square Roots

A square root of a number is one of its two equal factors. In simpler terms, the square root of \(a\) is a number that, when multiplied by itself, gives \(a\). Square roots are represented using the radical sign \(\sqrt{}\).
For instance, \(\sqrt{4}\) is 2, because \(2 \times 2 = 4\). Square roots are a type of radical, and they are specifically the inverse of squaring a number. When you encounter \(4^{\frac{1}{2}}\), you’re essentially finding \(\sqrt{4}\).
Here’s how it works:

• Imagine you have a square with an area of 4 square units. The side length of this square is the square root, which is 2.
• This is because a square with side length 2 has an area of 4 (since \(2 \times 2 = 4\)).

Square roots help us solve various types of equations and are useful in geometry and algebra.

Simplifying to the Simplest Form

Expressing numbers in their simplest form means reducing them to their most concise version without changing their value. A rational number is expressed as \(\frac{a}{b}\), where \(a\) and \(b\) are integers and \(b\) is not zero.
In the exercise, converting \(4^{\frac{1}{2}}\) resulted in \(2\), which is already a whole number. But \(2\) can be expressed as a rational number: \(\frac{2}{1}\). The fraction \(\frac{2}{1}\) is in its simplest form because there are no common factors to cancel out in the numerator and denominator.
When simplifying expressions:

• Identify if any common factors exist in the numerator and denominator.
• Ensure the greatest common divisor is \(1\), meaning it cannot be simplified further.
• Check to see if the result is already in its simplest form, as in the case of a whole number.

Doing this helps facilitate easier calculations in future arithmetic or algebraic operations.