Question

Evaluate the expressions in Exercises \(1-4\) without using a calculator. $$ 4^{1 / 2} $$

Short answer

Answer: The value of the expression \(4^{1/2}\) is \(2\).

Step-by-step solution

Analyze the exponent

We are given the expression \(4^{1/2}\). The exponent \(1/2\) represents a square root operation. So the expression can be rewritten as \(\sqrt{4}\).

Find the square root

Now, we need to find the square root of \(4\). Recall that a number's square root is the number that, when multiplied by itself, gives the original number. In this case, the number is \(4\). Since \(2\times2=4\), the square root of \(4\) is \(2\).

Write the final answer

We have found that \(\sqrt{4}=2\). So the evaluated expression \(4^{1/2}\) is also equal to \(2\). The final answer is \(2\).

Key concepts

Understanding Square Roots

Square roots are a fundamental concept in mathematics that allow us to reverse the process of squaring a number. In simple terms, finding the square root of a number means finding a value that, when multiplied by itself, results in the original number. For example:

• The square root of 9 is 3 because when 3 is multiplied by itself (3 × 3 = 9), it equals 9.
• Similarly, the square root of 16 is 4, since 4 × 4 = 16.

The notation for square root is \( \sqrt{} \). Thus, \( \sqrt{4} = 2 \), as 2 multiplied by 2 equals 4. Recognizing that an exponent of \( \frac{1}{2} \) is equivalent to taking a square root can simplify many expressions. This knowledge is especially useful when dealing with exponents involving fractions.

Multiplication Basics

Multiplication is one of the core arithmetic operations that combines groups of equal size. It's a shortcut for adding the same number several times. Think of multiplication as creating multiple piles of the same size. For instance,

• with 2 × 3 , you place two piles of three objects: 3 + 3 = 6.
• Similarly, 5 × 4 gives you five piles of four objects: 4 + 4 + 4 + 4 + 4 = 20.

This operation is commutative, meaning a × b is the same as b × a , and it makes solving complex problems more efficient. In the context of square roots, multiplication comes into play when you multiply a number by itself to verify it reverses the process of squaring. For example, to check the square root of 4, multiply 2 by 2: 2 × 2 = 4 .

Evaluating Expressions

Evaluating expressions is about finding the numerical value of mathematical statements. When you encounter an expression like \( 4^{1/2} \), you're tasked with simplifying it to find its value. Follow the order of operations, known as PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction), to tackle these problems systematically.In the expression \( 4^{1/2} \), the exponent indicates that you are dealing with a square root (\( \sqrt{4} \)). Once the expression is rewritten in a simpler form, you can solve it step by step. Find the square root first since it appears as an exponent operation. By applying these steps correctly, you're able to conclude that \( 4^{1/2} = 2 \). Evaluating expressions correctly aids in solving broader arithmetic and algebraic problems, ensuring accuracy in your mathematical computations.