Problem 78 - College Algebra

Question

In Exercises \(77-84,\) evaluate each expression without using a calculator. $$121^{1 / 2}$$

Step-by-Step Solution

Answer

The value of the expression \(121^{1/2}\) is 11.

Step 1: Understand the power '1/2'

The first step is to understand that any number raised to the power 1/2 is the same as taking the square root of that number. In this case, \(121^{1/2}\) is the same as finding \(\sqrt{121}\).

Step 2: Find the square root

The aim is now to find out a number which when squared gives 121. The numbers 1 to 10 can be considered because squaring anything above 10 will give a number greater than 121. Recall that 11 squared equals 121. Therefore, \(\sqrt{121}\) is 11.

Step 3: Write the final answer

The final answer after evaluating \(121^{1/2}\) or \(\sqrt{121}\) is found to be 11.

Key Concepts

Powers and Exponents

Powers and exponents are an essential mathematical tool that allows us to express repeated multiplication in a compact form. If you have a number, say "a," raised to the power of "n," it is written as \(a^n\). This tells you to multiply the number "a" by itself "n" times. For example, \(2^3\) means you multiply 2 by itself three times: \(2 \times 2 \times 2 = 8\).

Exponents have specific properties that simplify calculations:

• Product of Powers: \(a^m \times a^n = a^{m+n}\)
• Quotient of Powers: \(a^m / a^n = a^{m-n}\)
• Power of a Power: \((a^m)^n = a^{m\times n}\)

Understanding these properties can make complex problems more manageable. In our example with \(121^{1/2}\), recognizing that the exponent 1/2 represents a specific operation greatly simplifies the computation.

Rational Exponents

Rational exponents extend the concept of powers and provide a bridge to roots, like square roots or cube roots. When you see an expression like \(a^{1/n}\), what you're really seeing is the nth root of "a." So, \(a^{1/2}\) means you're looking for the square root of "a."

For example, let's evaluate \(121^{1/2}\). Here, the rational exponent is 1/2, which corresponds to the square root. You want to find that number, which when squared (multiplied by itself) returns the original number, 121. Calculating \(\sqrt{121}\), we identify that 11 multiplied by 11 equals 121, hence \(121^{1/2} = 11\).

This concept makes dealing with roots and fractional exponents straightforward since a power raises or lowers the complexity of a number systematically.

Evaluating Expressions

Evaluating expressions involves finding the value of an expression based on the operators and values it contains. When evaluating expressions with rational exponents or other operations, it's crucial to interpret these correctly based on mathematical rules.

Consider the expression \(121^{1/2}\), which involves evaluating a rational exponent. The initial step is to recognize that \(1/2\) as an exponent equates to finding a square root of 121. Recall the steps associated with this process:

• Identify the Operation: Understand that the exponent denotes a root.
• Execute the Operation: Find the precise root. Here, identify \(\sqrt{121}\) as 11.
• Write Your Result: Conclude the process by clearly stating your final answer as 11.

Each evaluation should be methodically approached for clear and accurate solutions to complex expressions.