Problem 83 - College Algebra Essentials

Question

In Exercises \(83-90\), evaluate each expression without using a calculator. $$36^{\frac{1}{2}}$$

Step-by-Step Solution

Answer

The value of \(36^{\frac{1}{2}}\) is 6.

Step 1: Understand the expression

The expression given, \(36^{\frac{1}{2}}\), represents the square root of 36. In general, \(x^{\frac{1}{2}}\) is equivalent to \(\sqrt{x}\) which is then the square root of x.

Step 2: Calculate the square root

To find the square root of a number, we look for a number which, when multiplied by itself, gives the original number. In this case, we know that \(6*6 = 36\). Therefore, the square root of 36 is 6.

Key Concepts

Rational Exponents

Rational exponents are a crucial part of simplifying expressions, especially when dealing with roots. Essentially, they provide an alternative way to express roots. A rational exponent is written in the form of \(a^{\frac{m}{n}}\), where \(a\) is the base, \(m\) is the numerator of the fraction representing the exponent, and \(n\) is the denominator. This notation translates to \(\sqrt[n]{a^m}\) or equivalently \((\sqrt[n]{a})^m\).

For example, in the exercise \(36^{\frac{1}{2}}\), the exponent \(\frac{1}{2}\) represents the square root, because the denominator 2 indicates a square root. Understanding this notation is essential as it provides a universal method to express not just square roots, but any nth root, and can significantly simplify complex algebraic expressions.

Rational exponents allow us to utilize the rules of exponents in equation solving and simplifying expressions. They open an efficient path to handle expressions that might otherwise be cumbersome to deal with directly using root symbols.

Exponentiation

Exponentiation is a mathematical operation that involves repeated multiplication of a base number. It's a fundamental concept that underlies many areas of algebra.

In an expression \(b^n\), \(b\) is the base and \(n\) is the exponent or power. The exponent indicates how many times the base is multiplied by itself. For example, \(3^4\) means \(3\times3\times3\times3\).

Key properties of exponents include:

• Product of Powers Rule: \(x^m \cdot x^n = x^{m+n}\)
• Power of a Power Rule: \((x^m)^n = x^{m \cdot n}\)
• Quotient of Powers Rule: \(\frac{x^m}{x^n} = x^{m-n}\) when \(xeq0\)
• Power of a Product Rule: \((xy)^n = x^n \cdot y^n\)
• Zero Exponent Rule: \(x^0 = 1\) for any non-zero \(x\)

Understanding these rules is vital for simplifying expressions and solving algebraic problems that involve exponents. In the context of the square root operation, recognizing it as a form of exponentiation (specifically \(x^{\frac{1}{2}}\)) illustrates how closely tied the operations of taking a square root and raising to a power are.

Algebraic Expressions

An algebraic expression is a combination of numbers, variables, and operations. These can be as simple as \(x + 2\) or as complex as \(3x^2 - 5x + 7\). They are essential in representing mathematical relationships and solving equations effectively.

Expressions are classified into different types, depending on the operations involved and the components.

• Polynomial Expressions: These have terms composed of variables raised to whole number exponents. Example: \(4x^3 - x^2 + 3\).
• Rational Expressions: These involve fractions where the numerator and/or the denominator are polynomials. Example: \(\frac{x^2 + 2x + 1}{x-1}\).
• Radical Expressions: These involve roots, such as square roots or cube roots, which can be rewritten using rational exponents. Example: \(\sqrt{x} = x^{\frac{1}{2}}\).

Understanding how to manipulate these expressions using techniques like combining like terms, factoring, and applying exponent rules is essential for solving algebraic equations and simplifying complex mathematical problems. The use of rational exponents, as demonstrated in the expression \(36^{\frac{1}{2}}\), highlights the versatility of algebraic expressions in representing different mathematical ideas efficiently.