Problem 6 - Algebra 1: Concepts and Skills

Question

Evaluate the expression without using a calculator. $$ 121^{1 / 2} $$

Step-by-Step Solution

Answer

The square root of 121 is 11 and -11

Step 1 : Understand the square root notation

The expression 121^{1 / 2} denotes the square root of 121. This is because any number to the power of 1/2 is the square root of that number.

Step 2 : Evaluate the square root

The square root of 121 is the number that, when multiplied by itself, results in 121. The numbers 11 and -11 are such numbers since \(11^2 = 121\) and \((-11)^2 = 121\)

Key Concepts

Exponentiation

Exponentiation refers to the process of raising a number, called a base, to the power of an exponent. The exponent tells us how many times the base is multiplied by itself. When you see something like \( a^n \), \( a \) is the base and \( n \) is the exponent.
For example, in \( 2^3 \), 2 is multiplied by itself three times: \( 2 \times 2 \times 2 = 8 \). This means that \( 2^3 = 8 \).

It's essential to understand that exponentiation is a key operation in mathematics. When the exponent is a fraction, it introduces us to roots, such as the square root or cube root. When an exponent is \( 1/2 \), it signifies the square root of the base. This crucial concept is what you encounter in expressions like \( 121^{1/2} \).
Understanding exponentiation lays the foundation for further mathematical concepts and helps in solving various mathematical problems efficiently.

Radical Notation

Radical notation is used to express roots of numbers. The most common type is the square root, symbolized as \( \sqrt{} \). In general terms, the radical notation \( \sqrt[n]{a} \) represents the \( n \)-th root of \( a \).

For example, \( \sqrt{121} \) indicates the square root of 121, which is 11, because \( 11 \times 11 = 121 \). Similarly, \( \sqrt[3]{8} \) would represent the cube root. Roots are essentially the inverse operations of exponentiation.
A key aspect of exponential and radical expressions is converting between them. For instance, \( a^{1/2} \) is the same as \( \sqrt{a} \). This dual representation provides flexibility in how equations are written and solved. Recognizing this relationship helps simplify complex mathematical tasks and enhances your problem-solving skills.

Number Operations

Number operations include basic arithmetic operations such as addition, subtraction, multiplication, and division. These form the backbone of mathematics and are crucial for understanding more complex concepts.

A key part of number operations in dealing with expressions like \( 121^{1/2} \) is finding the number that satisfies the condition of multiplication by itself: a square root. In this case, the operation needed is recognizing which number multiplies by itself to yield 121.

• For addition, consider \( a + b \)
• For subtraction, consider \( a - b \)
• For multiplication, consider \( a \times b \)
• For division, consider \( a \div b \)

Recognizing when and how to use these operations is vital. In our exercise, finding the number that when squared equals 121 involves multiplication, one of core number operations. Therefore, knowing basic arithmetic sets the stage for understanding and solving mathematical problems effectively.