Problem 77 - College Algebra

Question

In Exercises \(77-84,\) evaluate each expression without using a calculator. $$36^{1 / 2}$$

Step-by-Step Solution

Answer

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

Step 1: Understand the Problem

The operation \(36^{1 / 2}\) is a square root operation, since \(a^{1 / 2}\) always means the square root of \(a\).

Step 2: Perform the Square Root Operation

To perform the square root operation, it's necessary to think of a number which, when squared, gives 36. The number that gives 36 when squared is 6. So the square root of 36 is 6.

Step 3: State the Final Answer

Therefore, the value of the expression \(36^{1 / 2}\) is 6.

Key Concepts

Exponentiation

Exponentiation is a mathematical operation used to represent repeated multiplication of a number. In the simplest terms, if you have a number, called the base, and an exponent, the operation tells you to multiply the base by itself as many times as the exponent indicates. For example, in the expression \(a^n\), \(a\) is the base, and \(n\) is the exponent.

When the exponent is 1/2, as in \(36^{1/2}\), it refers to the square root of the base. This is because any number raised to the power of 1/2 is mathematically equivalent to taking the square root of that number. For square roots, we are looking for a number that when squared (multiplied by itself), returns the original base number.

• If the expression is \(9^{1/2}\), the result is 3, because \(3^2 = 9\).
• For \(25^{1/2}\), the result is 5, because \(5^2 = 25\).

Understanding how exponentiation works, especially when dealing with fractional exponents, is key to mastering square roots and other radical expressions.

Evaluating Expressions

Evaluating expressions involves substituting variables or constants, then performing mathematical operations according to the given operators. The goal is to simplify the expression to find the solution or value.

For our problem, evaluating the expression \(36^{1/2}\) means finding the value of this mathematical operation as given. Since exponentiation with a 1/2 denotes a square root, we look for a number that when squared equals 36. Through our knowledge of squares, we recognize that 6 is the number that multiplies by itself to equal 36, thus simplifying the expression to 6.

• Start by recognizing the operation involved: in this case, exponentiation implies a square root.
• Identify which number squared will yield the base number. For 36, this number is 6.
• State the evaluated answer: \(36^{1/2} = 6\).

Evaluating expressions properly requires understanding the operations, following order of operations (PEMDAS/BODMAS), and sometimes rewriting expressions for clarity.

Algebra Equations

Algebra equations involve solving for unknown values using operations like addition, subtraction, multiplication, division, and exponentiation.

Although our problem does not directly involve solving for an unknown, the concept still applies to understanding expressions like \(36^{1/2}\). If this were part of an equation \(x = 36^{1/2}\), you would solve it by determining that \(x = 6\).

• Recognize the relationship between the components of the equation.
• Use inverse operations to isolate variables when solving. For example, if an equation involved \(x^2 = 36\), taking the square root would help solve for \(x\).
• Check your work by substituting back to verify solutions.

Understanding algebraic equations and how to manipulate expressions is foundational for solving complex problems in mathematics. It helps you identify the steps needed to find unknowns based on given information.