Problem 17 - Applied Calculus

Question

Evaluate each expression without using a calculator. $$ 25^{1 / 2} $$

Step-by-Step Solution

Answer

5

Step 1: Identify the Expression

The given expression is \(25^{1/2}\). This can be interpreted as the square root of 25. The symbol \(^{1/2}\) is another way of expressing the square root.

Step 2: Recall the Definition of Square Root

The square root of a number \(x\) is a value \(y\) such that \(y^2 = x\). Therefore, to find \(25^{1/2}\), we need to determine what number \(y\) satisfies the equation \(y^2 = 25\).

Step 3: Find the Square Root of 25

The number whose square is 25 is 5, since \(5^2 = 25\). Hence, the square root of 25 is 5.

Key Concepts

Exponents

Exponents are mathematical expressions that denote repeated multiplication of a number by itself. When you see a number written with an exponent, it's telling you how many times to multiply the base (the large number) by itself. For example, in the expression\[ a^b \]\( a \) is the base and \( b \) is the exponent. If \( b = 2 \), then you multiply \( a \) by itself: \( a \times a \).

• An exponent of 2 is often referred to as "squared."
• An exponent of 3 is called "cubed."
• An exponent of 1/2, as in \( 25^{1/2} \), signifies the square root of the base.

Understanding exponents is crucial because they simplify expressions and calculations, especially when dealing with large numbers. Exponents are a key part of many areas of mathematics, including algebra and calculus.

Radicals

A radical, often symbolized with the square root sign \( \sqrt{} \), represents the root of a number. The most common radical is the square root, which finds what number multiplied by itself gives the original number. For instance, \( \sqrt{25} \) asks "What number squared equals 25?", and the answer is 5.
Radicals can express not just square roots, but cube roots, fourth roots, and so on. These are represented by a small number above the \( \sqrt{} \) symbol called the "index." Without an index, the radical is usually assumed to be a square root.

• Square roots: \( \sqrt{9} = 3 \) because \( 3^2 = 9 \)
• Cube roots: \( \sqrt[3]{8} = 2 \) because \( 2^3 = 8 \)
• Higher roots follow the same logic, but with the index indicating which root.

Radicals are important because they help us solve equations that involve roots and make sense of numbers that aren't easily squared or cubed.

Simplifying Expressions

Simplifying expressions involves breaking down complex equations into simpler forms that are easier to understand or solve. This often includes working with exponents and radicals.
In expressions involving exponents like \( 25^{1/2} \), simplifying means interpreting the expression in a clearer way—here it represents \( \sqrt{25} \). This is because an exponent of 1/2 translates directly to a square root.
Simplification involves several skills:

• Identifying equivalent expressions, such as recognizing that \( x^{a/b} \) is the \( b \)th root of \( x^a \).
• Reducing fractions within exponents or radicals where possible.
• Performing operations in a clear logical sequence.

By simplifying expressions, we make equations more manageable and easier to solve. This allows us to effectively tackle complex math problems and understand the relationships between numbers.