Problem 82 - Algebra 1

Question

Simplify the radical expression. $$\frac{1}{4} \sqrt{64}$$

Step-by-Step Solution

Answer

The simplified form of the radical expression \(\frac{1}{4} \sqrt{64}\) is \(2\).

Step 1: Calculate the Square Root

Start by calculating the square root of 64, which is \(8\). So, the expression now becomes \(\frac{1}{4} * 8\).

Step 2: Perform the Multiplication

Multiply \(\frac{1}{4}\) by \(8\). This equals \(2\). So, the original expression \(\frac{1}{4} \sqrt{64}\) simplifies to \(2\).

Key Concepts

Square Root Calculation

Understanding square roots is essential in simplifying radical expressions. A square root of a number is a value that, when multiplied by itself, gives the original number. In this exercise, we need to find the square root of 64. Since 8 multiplied by 8 equals 64, the square root of 64 is 8. Written in mathematical terms, this is expressed as:

• \( \sqrt{64} = 8 \)

Calculating square roots can be easy when dealing with perfect squares like 64. However, for non-perfect squares, you might need a calculator or estimate using methods such as prime factorization. By mastering these techniques, you'll be able to tackle any square root challenge!

Basic Multiplication

Once you've calculated the square root, it's time to multiply it by other numbers or fractions in the expression. In this exercise, we have \( \frac{1}{4} \times 8 \). To multiply a fraction by a whole number:

• Multiply the numerator of the fraction by the whole number.
• Divide the result by the denominator of the fraction.

Here, the numerator \(1\) multiplied by \(8\) gives us \(8\), and then dividing by the denominator \(4\) results in \(2\):

• \( \frac{8}{4} = 2 \)

Understanding these basic multiplication steps helps simplify expressions accurately. It's a fundamental skill not just in algebra but in many areas of mathematics.

Algebraic Simplification

Simplifying algebraic expressions involves performing operations to reduce them to their simplest form. We start with the original expression \( \frac{1}{4} \sqrt{64} \). By first calculating the square root and then performing multiplication, we have:

• Calculated \( \sqrt{64} = 8 \)
• Multiplied to get \( \frac{1}{4} \times 8 = 2 \)

The final simplified expression is \(2\). Simplification helps in reducing complexity and making expressions easier to work with. Whether you're dealing with numbers or variables, these techniques are crucial for solving problems efficiently in algebra.