Problem 14 - A Graphical Approach to College Algebra

Question

Simplify each expression. Assume that all variables represent positive real numbers. $$25^{1 / 2}$$

Step-by-Step Solution

Answer

The simplified expression is 5.

Step 1: Understand the notation

The expression is given as \(25^{1/2}\). Here, the exponent \(1/2\) means that we need to find the square root of \(25\).

Step 2: Rewrite the expression

The notation \(25^{1/2}\) can be rewritten using the radical symbol as \(\sqrt{25}\).

Step 3: Evaluate the square root

The square root of \(25\) is the number that, when multiplied by itself, gives \(25\). Since \(5 \times 5 = 25\), we have \(\sqrt{25} = 5\).

Key Concepts

Exponents

Exponents are a way to express repeated multiplication of a number by itself. When you see a number or a variable with an exponent, it means you multiply the base number by itself as many times as the exponent indicates.
For example, in the expression \(a^n\):

• \(a\) is the base.
• \(n\) is the exponent.

If \(n = 2\), \(a^2 = a \times a\); if \(n = 3\), \(a^3 = a \times a \times a\), and so forth.
There are also special types of exponents that you should be aware of:

• \(a^0 = 1\): Any number raised to the power of zero is 1.
• \(a^1 = a\): Any number raised to the power of one is the number itself.
• Negative exponents: \(a^{-n} = \frac{1}{a^n}\): You take the reciprocal of the base raised to a positive exponent.
• Fractional exponents, like \(a^{1/2}\) which imply a root, meaning you take a root of the base.

Understanding exponents is crucial for simplifying expressions, including fractions and radicals.

Square Roots

Square roots are mathematical functions that find a number, which when multiplied by itself gives the original number.
It's denoted by the radical symbol \(\sqrt{}\). When dealing with square roots:

• \(\sqrt{4} = 2\), because \(2 \times 2 = 4\).
• \(\sqrt{9} = 3\), because \(3 \times 3 = 9\).

The idea is to reverse the process of squaring a number. When you see an expression like \(25^{1/2}\), it actually means you need to find the square root of 25.
You can rewrite \(25^{1/2}\) as \(\sqrt{25}\), which equals 5, since 5 times 5 is 25.
Remember, the square root asks: "What number multiplies by itself to reach this number?"
This concept is foundational when simplifying expressions, especially when dealing with exponents expressed as fractions.

Radicals

Radicals are expressions that include a root, such as a square root, a cube root, etc. You will often see them represented by the radical symbol \(\sqrt{}\), with the most common radicals being square roots and cube roots.
Let's break it down:

• The square root \(\sqrt{a}\) is the number that, when squared, returns \(a\).
• The cube root \(\sqrt[3]{a}\) is the number which, when cubed, gives \(a\).
• Higher roots follow the same principle, \(a^{1/n}\) represents the nth root.

When simplifying expressions with radicals, especially when tied with other algebraic operations, look to rewrite them in their simplest form. For example:
1. If you have \(\sqrt{64}\), think: what number multiplied by itself equals 64? The answer is 8.
2. With \(\sqrt[3]{27}\), consider: what number multiplied thrice is 27? This is 3.
Understanding radicals will help you simplify complex expressions and solve equations efficiently, especially in algebraic contexts.