Problem 92 - Algebra and Trigonometry Real Mathematics, Real People

Question

Think About It In Exercises \(89-92,\) place the correct symbol \(( < \text { or } > )\) between the two numbers. $$4^{1 / 2} \quad\left(\frac{1}{2}\right)^{4}$$

Step-by-Step Solution

Answer

The correct symbol to place between the two numbers is '>', because 2 > \(\frac{1}{16}\)

Step 1: Compute the square root of 4

In mathematics, the square root of a number is a value that, when multiplied by itself, gives the original number. Therefore, we have \(4^{1 / 2} = 2\).

Step 2: Compute the fourth power of 1/2

In mathematics, the power (or exponent) of a number indicates the number of times the base is multiplied by itself. For example, \(\left(\frac{1}{2}\right)^{4}\) means \( \frac{1}{2} * \frac{1}{2} * \frac{1}{2} * \frac{1}{2} = \frac{1}{16}\) .

Step 3: Compare the two numbers

We now have both numbers: 2 and \(\frac{1}{16}\). It is clear that 2 is greater than \(\frac{1}{16}\).

Key Concepts

Understanding Square Root Calculations

The process of finding the square root of a number involves identifying a value which, when multiplied by itself, equals the original number. This can often seem abstract, but it's actually quite a simple concept when broken down.

For example, as seen in the exercise above, the square root of 4, denoted as \(4^{1/2}\), is 2 because multiplying 2 by itself gives 4 (\(2 \times 2 = 4\)). This 'half' exponent represents the square root operation. It's essential to master understanding square roots since they are foundational in solving many mathematical problems, especially those involving quadratic equations and geometric calculations involving area.

Exponent Calculations Explained

When dealing with exponent calculations, you are working with a base number raised to a certain power. This represents the base number multiplied by itself a number of times equal to the exponent.

In the original exercise, the calculation \(\frac{1}{2}^4\) can be translated to \(\frac{1}{2} \times \frac{1}{2} \times \frac{1}{2} \times \frac{1}{2}\). This results in \(\frac{1}{16}\). The important thing to remember with exponents is that they follow specific rules, like the product rule, quotient rule, and power rule, all of which help simplify complex calculations. Understanding how to manipulate exponents is crucial in algebra and beyond as they appear across various areas of mathematics.

Inequalities in Mathematics

Inequalities are mathematical expressions that compare two values, showing if one is less than, greater than, or equal to the other. They are symbolized using '<' for less than, '>' for greater than, and '=' for equal to.

Examining the given exercise, we have to compare the results of the square root and exponent calculations. After computing both values (2 and \(\frac{1}{16}\)), we use an inequality to state that 2 is greater than \(\frac{1}{16}\). Understanding inequalities is fundamental in mathematical reasoning and appears in various contexts, from solving equations to analyzing functions and interpreting real-world situations.