Problem 13 - A Graphical Approach to College Algebra

Question

Simplify each expression. Assume that all variables represent positive real numbers. $$-4^{1 / 2}$$

Step-by-Step Solution

Answer

The simplified expression is \(-2\).

Step 1: Identify the Exponent

The expression \(-4^{1/2}\) contains a negative base \(-4\) with an exponent \(1/2\). The exponent indicates that we need to find the square root of the base.

Step 2: Simplify the Square Root

The expression \(-4^{1/2}\) is equivalent to \(-\sqrt{4}\). The square root of 4 is \(2\), so our expression simplifies to \(-2\).

Key Concepts

Exponentiation

Exponentiation is a mathematical operation involving numbers raised to a power. It allows us to express repeated multiplication in a simple format. For example, in the expression \[a^b\]\(a\) is the base, and \(b\) is the exponent. This indicates that \(a\) should be multiplied by itself \(b\) times.
When dealing with fractions as exponents, such as \[a^{1/2},\]the exponent indicates that you are looking for the square root of the base \(a\).
In our exercise, the negative sign is outside the exponential operation. This tells us that we are looking at the negative of the square root result.

• Fractions in exponents represent roots: \(a^{1/n}\) equals the \(n\)-th root of \(a\).
• The base can be any real number, including a negative value as shown in this problem.

Understanding these principles helps in efficiently tackling problems involving exponentiation.

Square Roots

Square roots are a special case of roots, often indicated by an exponent of \(1/2\). When you see \[x^{1/2}\]it means you need to find a number that, when multiplied by itself, gives \(x\).
"Finding the square root" can be simply thought of as the opposite of squaring a number. For instance, the square root of \(4\) is \(2\) because \(2 \times 2 = 4\).
In our example, we are asked to find the square root of \(4\). This yields \(2\), meaning that the expression simplifies directly to \[-2\].

• Square roots are positive by default, but here we apply a negative sign as indicated.
• Always consider the placement of negative signs in expressions.

Remember, understanding square roots is fundamental in simplifying expressions involving power of half.

Simplification

Simplification is the process of making an expression easier to understand or compute. It involves reducing expressions to their most concise form without changing their value.
In the original exercise, \(-4^{1/2}\)starts out seemingly complex but can be simplified to \[-2\]by following key steps. Here's how the simplification worked:

• Recognize the operation: Identify the type of power (here it's a square root).
• Break down the expression: Convert it into a simpler equivalent (\(-\sqrt{4}\)).
• Compute and apply any signs: Find the square root of \(4\) and apply the negative, resulting in \(-2\).

By breaking the expression into understandable parts, you achieve simplification. This helps make problems more manageable and reveals insights into the operations involved.