Problem 11 - Precalculus : Building Concepts and Connections

Question

Evaluate each expression to four decimal places using a calculator. $$3^{\sqrt{2}}$$

Step-by-Step Solution

Answer

The result of evaluating the expression \(3^{\sqrt{2}}\) and rounding it to four decimal places is 2.2796.

Step 1: Identify the Expression

The given mathematical expression is \(3^{\sqrt{2}}\).

Step 2: Apply the Square Root

We need to first understand that \(\sqrt{2}\) is the square root of 2, which will be evaluated first as per the order of operations (BIDMAS/BODMAS). Using a calculator, \(\sqrt{2}\) is approximately 1.4142.

Step 3: Evaluate the Power

Now that we have the value of \(\sqrt{2}\), we substitute it in the expression \(3^{\sqrt{2}}\) so we have \(3^{1.4142}\). Now use a calculator to compute this value.

Step 4: Round to Four Decimal Places

Once \(3^{1.4142}\) is calculated, the last step is to round this value to four decimal places, as requested by the exercise.

Key Concepts

Order of Operations

When evaluating mathematical expressions, it's important to follow a specific sequence known as the order of operations. This rule dictates the correct sequence in which to solve parts of the expression. Commonly remembered with the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction) or BIDMAS/BODMAS (Brackets, Indices or Orders, Division and Multiplication, Addition and Subtraction).

Understanding and applying this rule ensures that everyone solves expressions in the same way and gets the same results. For example, in the expression given in the exercise, \(3^{\sqrt{2}}\), the square root is computed first because it's considered as an exponent-related operation, according to the rule.

Square Root

The square root of a number is a value that, when multiplied by itself, gives the original number. Expressed as \(\sqrt{x}\), where x is the number, finding the square root is essential in many areas of mathematics. Calculators often have a dedicated square root function that gives an immediate result. For example, \(\sqrt{2}\) can be approximated to 1.4142.

Understanding square roots is vital for higher-level math such as geometry, algebra, and calculus, where roots are used to simplify expressions and solve equations.

Exponentiation

Exponentiation is an operation that involves raising a number called the base to the power of an exponent. It's written as \(b^n\), meaning the base \(b\) is multiplied by itself \(n\) times. Calculating exponentiation is straightforward with a calculator that has a power function, often represented as \(x^y\) or \(y^{\text{x}}\) button.

In \(3^{1.4142}\), we're raising 3 to the power of 1.4142, which involves a non-integer exponent. It is an example of how exponentiation can extend beyond simple whole numbers and become more complex.

Rounding Decimal Places

Rounding decimal places is a numerical process used to reduce the number of digits right of the decimal while maintaining a close approximation to the original value. It is often used for convenience, simplicity, or to align with a specific level of precision required.

The rule for rounding is straightforward: look at the digit after the last desired decimal place—if it is 5 or greater, round up the last wanted digit; otherwise, leave it as is. For instance, when rounding \(3^{1.4142}\) to four decimal places, you would consider the fifth decimal place to decide whether to round the fourth decimal place up or leave it the same.