Problem 8 - Precalculus: Functions and Graphs

Question

Express the number in the form \(a / b,\) where \(a\) and \(b\) are integers. $$9^{1 / 2}$$

Step-by-Step Solution

Answer

\(9^{1/2} = \frac{3}{1}\).

Step 1: Understand the Expression

We need to express the given number \(9^{1/2}\) in the form \(a/b\), where \(a\) and \(b\) are integers. The expression \(9^{1/2}\) is a way of saying "the square root of 9."

Step 2: Calculate the Square Root

To find \(9^{1/2}\), calculate the square root of 9. As 9 is a perfect square where \(3 \times 3 = 9\), the square root of 9 is 3. Thus, \(9^{1/2} = 3\).

Step 3: Express as a Fraction

We found that \(9^{1/2} = 3\). To express this in the form \(a/b\), recognize that 3 is equivalent to \(\frac{3}{1}\). Therefore, the expression in the form \(a/b\) is \(\frac{3}{1}\).

Key Concepts

Exponents

Exponents are an essential concept in precalculus that describe how many times a number, known as the base, is multiplied by itself. For example, in the expression \(9^{1/2}\), 9 is the base and \(1/2\) is the exponent. This particular exponent indicates a special operation, which is the square root.

Exponents are written in the form of \(a^b\), where \(a\) is the base and \(b\) is the exponent. The different forms of exponents can change the operation carried out on the base, such as squaring, cubing or finding a root.

Important points about exponents include:

• Positive integers: Indicate how many times to multiply the base by itself (e.g., \(3^2 = 3 \times 3 = 9\)).
• Fractional exponents: Represent roots (e.g., \(9^{1/2}\) represents the square root of 9).
• Zero exponents: Any non-zero base raised to the power of zero equals 1 (e.g., \(5^0 = 1\)).

Understanding exponents helps us simplify complex expressions and solve equations more easily.

Square Roots

Square roots are a type of radical expression that determine what number, when multiplied by itself, results in the given number. The square root of 9, symbolized as \(\sqrt{9}\), asks what number squared is equal to 9. In this case, the answer is 3, since \(3 \times 3 = 9\).

Calculating square roots is a common operation in mathematics, especially when dealing with perfect squares like 4, 9, and 16. Here are some important points about square roots:

• A square root can be thought of as the opposite of squaring a number.
• A perfect square is a number whose square root is an integer (e.g., 4, 9, 16).
• The square root symbol (\(\sqrt{}\)) helps identify these types of problems.

Knowing how to find square roots is crucial in simplifying expressions, solving quadratic equations, and other operations in algebra and precalculus.

Fractions

Fractions are used to represent parts of a whole. They consist of a numerator (the top number) and a denominator (the bottom number). In the case of expressions like \(\frac{3}{1}\), the fraction represents the whole number. The numerator is 3, and the denominator is 1, indicating that 3 is considered in its entirety.

In the context of the original problem where we found that \(9^{1/2} = 3\), expressing 3 as \(\frac{3}{1}\) aligns it with the fraction format \(a/b\). The fraction simplifies to the integer 3.

Key aspects of fractions include:

• Identifying parts of a whole and having infinite representations (e.g., \(\frac{3}{1}\) is equivalent to 3).
• Operations with fractions like addition, subtraction, multiplication, and division.
• Converting between improper fractions and mixed numbers.

Fractions are versatile and particularly important in expressing real-world quantities precisely.