Problem 5 - Algebra 2 and Trigonometry

Question

In \(3-37,\) express each power as a rational number in simplest form. $$ 100^{\frac{1}{2}} $$

Step-by-Step Solution

Answer

\(100^{\frac{1}{2}}\) simplifies to 10.

Step 1: Understand what the expression means

The expression \(100^{\frac{1}{2}}\) represents the square root of 100. In general, any expression of the form \(a^{\frac{1}{n}}\) is equivalent to the \(n\)-th root of \(a\). For this specific problem, \(n = 2\), indicating a square root.

Step 2: Calculate the square root

Find the square root of 100. The square root of a number \(x\) is a number \(y\) such that \(y \times y = x\). For this problem, the square root of 100 is 10, because \(10 \times 10 = 100\).

Step 3: Express as a rational number

Since the square root of 100 is 10, and 10 is a rational number (it can be expressed as \(\frac{10}{1}\)), the expression \(100^{\frac{1}{2}}\) is equivalent to the rational number 10.

Key Concepts

Exponentiation

Exponentiation is a mathematical operation involving two numbers: the base and the exponent. The base is the number that is multiplied, and the exponent tells us how many times the base is used as a factor. For example, in the expression \(b^n\), \(b\) is the base and \(n\) is the exponent. So, if \(b = 2\) and \(n = 3\), then \(2^3 = 2 \times 2 \times 2 = 8\).

In our problem, the expression \(100^{\frac{1}{2}}\) uses a fractional or rational exponent. This makes the process slightly different from whole-number exponents.

**Fractional Exponents**
Fractional exponents, like \(\frac{1}{2}\), represent roots. Specifically, \(a^{\frac{1}{n}}\) is the same as the \(n\)-th root of \(a\).

• For \(a^{\frac{1}{2}}\), it signifies the square root of \(a\).
• This allows complex roots to be handled in a simpler, unified form with other exponent rules.

Square Roots

The concept of square roots is fundamental when dealing with expressions that involve a rational exponent of \(\frac{1}{2}\). The square root of a number \(x\) is a value that, when multiplied by itself, equals \(x\).

For example, in our original exercise, we are tasked with finding \(100^{\frac{1}{2}}\). This is another way of asking what the square root of 100 is. We know that \(10 \times 10 = 100\), so the square root of 100 is 10.

**Properties of Square Roots**

• Square roots can be both positive and negative since both \(10\times10\) and \((-10)\times(-10)\) equal 100.
• However, in most contexts, the positive square root is considered the principal square root.

Understanding square roots is essential for simplifying expressions with fractional exponents.

Simplifying Expressions

Simplifying expressions is a process of altering an expression to its simplest, most efficient form. This involves a variety of operations and rules. Simplification makes expressions easier to evaluate and compare.

In the context of our exercise, we started with \(100^{\frac{1}{2}}\) and simplified it to the rational number 10.

**Steps for Simplifying Expressions with Rational Exponents**

• Recognize the base and the exponent in the expression.
• Rewrite the expression using root notation if necessary, like turning \(a^{\frac{1}{2}}\) into \(\sqrt{a}\).
• Evaluate the root, such as calculating \(\sqrt{100} = 10\).
• Express the result as a rational number, for instance, \(\frac{10}{1}\).

By simplifying the expression \(100^{\frac{1}{2}}\) to 10, we greatly reduce its complexity and make it easier to handle in equations or further operations.