Problem 2 - Algebra and Trigonometry

Question

Using radicals, we can write \(5^{1 / 2}\) as _________

Step-by-Step Solution

Answer

\(\sqrt{5}\)

Step 1: Understand the Expression

We start by examining the given expression \(5^{1/2}\). This exponent of \(1/2\) indicates that we are dealing with a power that can be rewritten using radicals.

Step 2: Recognizing the Relationship Between Exponents and Radicals

Recall that an expression of the form \(a^{1/b}\) can be rewritten using radicals as \(\sqrt[b]{a}\), where \(b\) is the root degree. For our given expression, \(5^{1/2}\), \(b\) is 2.

Step 3: Rewrite the Expression as a Radical

Using the relationship we've identified, \(5^{1/2}\) can be rewritten as a square root. Since \(b = 2\) in our expression, we represent this as \(\sqrt{5}\).

Key Concepts

Exponents

Exponents are a way to express repeated multiplication compactly. For example, instead of writing \(5 \times 5\), we use \(5^2\). Here, 5 is the base, and 2 is the exponent, which tells us how many times the base is multiplied by itself.

There are important rules for handling exponents:

• Product of Powers: Add the exponents if the bases are the same, e.g., \(a^m \times a^n = a^{m+n}\).
• Power of a Power: Multiply the exponents, e.g., \((a^m)^n = a^{m \times n}\).
• Quotient of Powers: Subtract the exponents if the bases are the same, e.g., \(\frac{a^m}{a^n} = a^{m-n}\).

Understanding these properties helps simplify calculations and solve equations involving exponents.

When you see a fractional exponent, like \(5^{1/2}\), it introduces a connection with radicals, which we'll explore next.

Square Root

A square root is essentially the inverse operation of squaring a number. The square root of a number \(a\) is another number, which, when multiplied by itself, results in \(a\). For example, the square root of 25 is 5 because \(5 \times 5 = 25\).

When we talk about \(\sqrt{a}\), it is equivalent to \(a^{1/2}\). This means that the operation of finding a square root is similar to raising a number to the exponent of \(1/2\). Let's look at some key points about square roots:

• They always have two possible values: a positive and a negative, since both \(5^2\) and \((-5)^2\) result in 25.
• Only non-negative numbers have real square roots to simplify, as real square roots can't be computed for negative numbers within the real number system.
• Simplifying radicals often involves finding perfect squares that are factors of the number under the radical sign.

By recognizing how exponents relate to square roots, we can write expressions like \(5^{1/2}\) as \(\sqrt{5}\).

Algebraic Expressions

Algebraic expressions involve numbers, variables, and different mathematical operations. They can include exponents and radicals, making them versatile in representing mathematical relationships.

Here are a few components you might encounter in algebraic expressions:

• Constants: Numbers without variables, like 3 or -5.
• Variables: Symbols that represent unknown values, often \(x, y, z\).
• Coefficients: Numbers multiplied by variables, e.g., in \(3x\), 3 is the coefficient.
• Operators and Operations: Include addition, subtraction, multiplication, division, and exponents, all of which organize the behavior and relationships within the expression.

Simplifying algebraic expressions can involve applying the laws of exponents or rewriting terms using radicals. Working with radicals in algebraic expressions is particularly useful in solving equations, as it allows conversion between radical form and exponential form.

Understanding how radical expressions and exponents are applied in algebra helps solve more complex mathematical problems, offering a bridge between pure arithmetic and more abstract numerical ideas.