Problem 84 - Algebra and Trigonometry

Question

Evaluate each expression without using a calculator. $$ 121^{\frac{1}{2}} $$

Step-by-Step Solution

Answer

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

Step 1: Understand the notation

The expression \( 121^{\frac{1}{2}} \) is the same as the square root of 121, represented as \( \sqrt{121} \). This is based on the property that the square root of a number can also be represented as raising that number to the power of \(\frac{1}{2}\).

Step 2: Recall perfect squares

The number 121 is a perfect square, which means it is the product of some integer with itself. In this case, the integer is 11 because \( 11*11 = 121 \).

Step 3: Apply the square root

Thus, the square root of 121 is 11, so \( \sqrt{121}=11 \) or \( 121^{\frac{1}{2}}=11 \).

Key Concepts

Square Root

The concept of a square root is fundamental in mathematics, especially when working with exponents and powers. The square root of a number is a value that, when multiplied by itself, gives the original number. For example, the square root of 25 is 5, because 5 multiplied by 5 equals 25. This can be symbolized as \( \sqrt{25} = 5 \).
The expression \( 121^{\frac{1}{2}} \) is another way of representing the square root of 121. In general, raising a number to the power of \( \frac{1}{2} \) is equivalent to finding its square root. This is because the exponent \( \frac{1}{2} \) and the square root operation are directly related. Understanding this relationship helps simplify complex mathematical expressions.

Perfect Squares

Perfect squares are numbers that result from a whole number multiplied by itself. They play a crucial role in simplifying expressions and solving equations. Examples include 1, 4, 9, 16, 25, where each of these is a result of an integer squared:

• \( 1^2 = 1 \)
• \( 2^2 = 4 \)
• \( 3^2 = 9 \)
• \( 4^2 = 16 \)
• \( 5^2 = 25 \)

In the provided exercise, 121 is identified as a perfect square because it equals \( 11^2 \), or 11 multiplied by 11. Recognizing perfect squares quickly helps evaluate expressions involving square roots efficiently.

Evaluating Expressions

Evaluating expressions involves simplifying them to find their value. This process often requires a good understanding of mathematical concepts such as exponents and roots. Let's look at how this applies to an expression like \( 121^{\frac{1}{2}} \).
Firstly, rewrite the expression using the equivalent square root notation, \( \sqrt{121} \). Recognize that 121 is a perfect square, which simplifies finding the square root to basic multiplication knowledge: 11 times 11 equals 121.
Thus, the expression evaluates to 11. The key to efficiently evaluating expressions like this one is to identify perfect squares early on, which allows for rapid computation without complex calculations or tools.