Problem 2 - Algebra for College Students

Question

Evaluate each numerical expression. \(64^{\frac{1}{2}}\)

Step-by-Step Solution

Answer

8

Step 1: Understanding the Expression

The expression given is a power expression, specifically a fractional exponent. When you see \(64^{\frac{1}{2}}\), it actually represents the square root of 64.

Step 2: Rewriting the Expression

Rewrite \(64^{\frac{1}{2}}\) using the radical notation. This can be expressed as \(\sqrt{64}\). Since taking a number to the power of \(\frac{1}{2}\) means finding the square root, \(64^{\frac{1}{2}} = \sqrt{64}\).

Step 3: Calculating the Square Root

Now compute the square root of 64. The square root of a number is a value that, when multiplied by itself, gives the original number. \(\sqrt{64} = 8\), because \(8 \times 8 = 64\).

Step 4: Concluding the Calculation

After calculating the square root in step 3, we find that \(64^{\frac{1}{2}} = 8\). Therefore, the value of the expression is 8.

Key Concepts

Square Roots

The concept of a square root is foundational in mathematics. When we talk about square roots, we are referring to finding a number which, when multiplied by itself, yields the original number. For example, when you encounter the square root symbol \( \sqrt{} \), it asks you to determine the number that does this squared operation.
For instance, the square root of 64 is determined by asking, "What number multiplied by itself equals 64?" In this case, the answer is 8, as \( 8 \times 8 = 64 \).
This simple operation allows us to solve problems easily but becomes vital in more advanced mathematical concepts such as solving quadratic equations and dealing with complex numbers.
Remember:

• The square root of a perfect square is always an integer.
• Every positive real number has two square roots: one positive and one negative. But in standard practice, the principal square root is taken as positive.

Radical Notation

Radical notation is a way of representing roots using the radical symbol (\( \sqrt{} \)). This notation helps simplify expressions that involve roots in various forms.
In fractional exponents, radical notation is useful for rewriting expressions to analyze and solve them more easily. For example, the expression \( 64^{\frac{1}{2}} \) can be simplified using radical notation to \( \sqrt{64} \). This makes it clear that we're dealing with the square root of 64.
Here are some points to understand about radical notation:

• The index of the radical indicates the root you are taking. In the case of square roots, the index is 2, though it is often omitted.
• Radicals can extend beyond square roots to cube roots \( \sqrt[3]{} \), fourth roots \( \sqrt[4]{} \), and so on.
• Using radical notation helps clarify mathematical operations, particularly in expressions involving multiple steps.

Understanding radical notation is crucial as it forms the basis for learning about more complex root expressions and helps in performing arithmetic involving roots and powers.

Evaluating Expressions

Evaluating expressions, especially those involving fractional exponents, is a fundamental skill in algebra and higher-level math. Evaluating an expression simply means calculating its value.
For the expression \( 64^{\frac{1}{2}} \), knowing how to handle fractional exponents is essential. These exponents tell us to perform an operation involving a root. Specifically, \( \frac{1}{2} \) indicates taking the square root, turning \( 64^{\frac{1}{2}} \) into \( \sqrt{64} \).
Here's how you can evaluate such expressions efficiently:

• Understand what the fractional exponent denotes—whether it’s indicating a square root or any other root.
• Convert the expression to radical notation to simplify the evaluation process.
• Perform the necessary calculations, in this case, recognizing that \( \sqrt{64} = 8 \).

This method allows you to tackle more complex problems methodically, ensuring an understanding of how and why certain operations are performed. In more advanced mathematics, this technique is invaluable for simplifying and solving equations involving fractional and negative exponents.