Problem 35 - Algebra 2

Question

Simplify each number. $$4^{1.5}$$

Step-by-Step Solution

Answer

So, \(4^{1.5}\) simplifies to \(2\).

Step 1: Understand the fractional exponent

The first thing to understand in this problem is that a fractional exponent can be written as a root. In this case, \(4^{1.5}\) can be written as \(\sqrt{4^{1.5}}\).

Step 2: Rewrite as root

Now we can rewrite \(4^{1.5}\) as \(\sqrt[2]{4^{1.5}}\) which is equivalent to \(\sqrt[2]{4}^1.5\) as the root of 2 is the square root.

Step 3: Simplify the square root

We know that the square root of \(4\) is \(2\), so this simplifies to \(2^{1.5}\).

Step 4: Simplify the exponent

Now, \(2^{1.5}\) can be written as \(\sqrt[2]{2}^1\), which is just \(2\).

Key Concepts

Fractional Exponents

When dealing with fractional exponents, it's crucial to understand how they relate to roots. In mathematics, a fractional exponent indicates both an exponentiation and a root. For instance, the expression \(a^{m/n}\) suggests that you take the nth root of \(a\) and then raise the result to the power of \(m\). This is equivalent to \((\sqrt[n]{a})^m\).

• A whole number in the denominator indicates the root. For example, a denominator of 2 refers to the square root.
• The numerator tells you to what power the base is raised after taking the root.

Fractional exponents provide a versatile way to represent expressions that involve both powers and roots in a simplified notation. This dual representation helps in solving and simplifying expressions systematically.

Square Roots

Square roots are a type of root where the value is squared (multiplied by itself) to return to the original value. For example, the square root of 4, written as \(\sqrt{4}\), is 2 because \(2 \times 2 = 4\).

• Remember that the square root of a number finds a value that, when multiplied by itself, results in the original number.
• It helps in simplifying expressions with exponents, especially when the exponent is a fraction involving 2, such as in \(4^{1.5}\).

Understanding square roots is essential when working with fractional exponents as it allows you to easily transform and simplify expressions.

Simplification of Expressions

Simplification involves reducing an expression into its simplest form, where further reductions or transformations are not possible. This process involves breaking down an expression by performing operations such as root extractions and power reductions.

• In the case of fractional exponents, transforming them into roots helps in simplifying the expression.
• By systematically applying the rules of exponents and roots, you transform complex expressions into simpler equivalents.
• Understanding these rules makes it easier to handle more complicated expressions in mathematics.

In our example \(4^{1.5}\), rewriting it using roots helps simplify it gradually: \(\sqrt{4^{1.5}} = \sqrt[2]{4^{1.5}}\), and then determining \(2^{1.5}\) as \((\sqrt[2]{2})^1\), which simplifies to \(2\). Using these simplification techniques makes handling mathematical expressions more manageable and less intimidating.