Problem 44 - Algebra A Combined Function

Question

Evaluate each exponential expression. $$ 49^{1 / 2} $$

Step-by-Step Solution

Answer

7

Step 1: Understanding the Expression

The expression we need to evaluate is \( 49^{1/2} \). Here, the exponent \( \frac{1}{2} \) represents the square root. Therefore, \( 49^{1/2} = \sqrt{49} \).

Step 2: Calculating the Square Root

Now we need to find the square root of 49. The square root of 49 is the number that, when multiplied by itself, equals 49.

Step 3: Identifying the Square Root

The number 7, when multiplied by itself (\(7 \times 7\)), equals 49. Therefore, \( \sqrt{49} = 7 \).

Step 4: Final Evaluation

Having identified that \( \sqrt{49} = 7 \), we conclude that \( 49^{1/2} = 7 \).

Key Concepts

Square Root

The concept of a square root is foundational in mathematics. A square root of a number is simply another number which, when squared, gives the original number. For instance, if you have the number 49, the square root would be 7, because \( 7 \times 7 = 49 \). Understanding square roots is crucial for evaluating expressions that involve fractional exponents.

• The square root symbol is \( \sqrt{} \).
• It asks "what number multiplied by itself equals the original number?"

Square roots simplify complex expressions and help in solving quadratic equations and other algebraic operations.

Evaluating Expressions

Evaluating expressions involves finding the value of an expression given certain values for the variables in it, or in our focus from a base number with its respective exponent. In our exercise, the expression \( 49^{1/2} \) needs evaluation. Initially, it's important to fully understand any mathematical notations present, as they guide you in determining the value.

• Identify the base and the exponent in the expression.
• Transform any fractional exponents into their root equivalents.

For this exercise, acknowledge the exponent \( 1/2 \) as a symbol for the square root, converting the expression into \( \sqrt{49} \), which simplifies to 7.

Exponents

Exponents denote how many times a base number is multiplied by itself. In our case, \( 49^{1/2} \) uses a fractional exponent of \( 1/2 \). Recognizing fractional exponents is essential.

• A positive exponent indicates repeated multiplication.
• A negative exponent implies division or reciprocal.
• A fractional exponent signifies roots; for example, \( 42^{1/2} \) signifies the square root of 42.

By understanding these variations, you can navigate through a wide range of algebraic situations efficiently.

Algebra

Algebra is about finding unknown variables through known values and mathematical operations. It frames a broad range of mathematical principles including the use of expressions with exponents.

• It utilizes symbols and numbers to express equations and relationships.
• Simplifies complex problems into solvable equations.
• Supports calculations such as solving for \( x \) in equations like \( x^2 = 49 \).

Understanding algebra opens up various mathematical vistas, enabling you to solve expressions like \( 49^{1/2} \) easily and prepares you for tackling more advanced math topics effectively.