Problem 83 - Algebra and Trigonometry

Question

Evaluate each expression without using a calculator. $$ 36^{\frac{1}{2}} $$

Step-by-Step Solution

Answer

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

Step 1: Understanding the Expression

The expression presented \(36^{\frac{1}{2}}\) is essentially asking for the square root of 36. In general, \(a^{\frac{1}{2}}\) refers to the square root of the number 'a'.

Step 2: Evaluating the Expression

The square root of 36 can be found knowing that it would be a number, when squared, gives 36. The numbers that fulfill this criteria are 6 and -6. Hence, the expression \(36^{\frac{1}{2}}\) evaluates to 6 and -6.

Key Concepts

Understanding the Square Root

The square root of a number is a value that, when multiplied by itself, gives the original number. It is represented by the symbol \( \sqrt{} \). For example, the square root of 36 is 6 because \( 6 \times 6 = 36 \). It's important to remember that every positive number actually has two square roots: one positive and one negative.

• Positive square root: \( \sqrt{36} = 6 \)
• Negative square root: \( -\sqrt{36} = -6 \)

This means that when you are asked to find the square root, like in the expression \( 36^{\frac{1}{2}} \), you consider both \( 6 \) and \( -6 \). Always checking both solutions can be quite helpful, especially in algebraic contexts.

Exploring Rational Exponents

Rational exponents are another way to express roots and powers in mathematics. They enable us to write roots like square roots and cube roots using exponents. A rational exponent such as \( a^{\frac{m}{n}} \) means "the \( n^{th} \) root of \( a \) raised to the \( m \)."
For instance, when we see \( a^{\frac{1}{2}} \), this means \( \sqrt{a} \). Therefore, \( 36^{\frac{1}{2}} \) equates to the square root of 36. This conversion from radical to exponent form makes it easier to perform algebraic operations.

• \( a^{\frac{1}{n}} \) is equivalent to \( \sqrt[n]{a} \)
• \( a^{\frac{m}{n}} \) can be interpreted as \( \left(\sqrt[n]{a}\right)^m \text{ or } \sqrt[n]{a^m} \)

Using rational exponents can simplify complex expressions and improve understanding of algebraic manipulations.

Algebraic Expressions Simplified

Algebraic expressions are combinations of variables, numbers, and operations. They often require simplification or evaluation based on given values or expressions, like the square root or rational exponents.
When dealing with algebraic expressions involving exponents, it’s essential to understand how multiplying, adding, or raising powers affect the expression. For example:

• \( x^a \times x^b = x^{a+b} \)
• \( \left(x^a\right)^b = x^{ab} \)
• \( \frac{x^a}{x^b} = x^{a-b} \)

This knowledge aids in breaking down and simplifying complex algebraic terms into more manageable sections. Recognizing that roots and powers can be interchangeably expressed using rational exponents unlocks more straightforward techniques for handling these algebraic scenarios.