Problem 68 - Algebra 2

Question

Simplify. $$ \sqrt{\frac{9}{36} x^{4}} $$

Step-by-Step Solution

Answer

\( \frac{1}{2} x^2 \)

Step 1: Simplify the Fraction Inside the Radicand

The given expression is \( \sqrt{\frac{9}{36} x^4} \). Begin by simplifying the fraction \( \frac{9}{36} \) under the square root. Divide both the numerator and the denominator by their greatest common divisor, which is 9. This simplifies to \( \frac{1}{4} \).

Step 2: Apply the Property of Square Roots to the Fraction

Now that we have \( \sqrt{\frac{1}{4} x^4} \), use the property \( \sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}} \). Apply this to get \( \frac{\sqrt{1}}{\sqrt{4}} x^2 \).

Step 3: Simplify the Square Root of Each Part

Calculate \( \sqrt{1} \) and \( \sqrt{4} \). Since \( \sqrt{1} = 1 \) and \( \sqrt{4} = 2 \), the expression simplifies to \( \frac{1}{2} x^2 \).

Step 4: Combine for the Simplified Result

Combine the factors to express the final simplified result: \( \frac{1}{2} x^2 \).

Key Concepts

Square Roots

The square root is a fundamental concept in algebra and math. It's like asking, "What number, when multiplied by itself, equals this number?" For example, the square root of 9 is 3, because 3 multiplied by 3 equals 9. This can be written mathematically as \( \sqrt{9} = 3 \). Square roots are particularly useful for simplifying expressions, especially when they're inside fractions or radical expressions.

When dealing with square roots in algebraic expressions, there are useful properties you can apply. One key property of square roots is that the square root of a fraction \( \sqrt{\frac{a}{b}} \) can be rewritten as \( \frac{\sqrt{a}}{\sqrt{b}} \). This means you take the square root of the numerator and the square root of the denominator separately. This property often helps to simplify expressions more easily.

In our example, \( \sqrt{\frac{1}{4}} \) breaks down to \( \frac{\sqrt{1}}{\sqrt{4}} \), which equals \( \frac{1}{2} \) since \( \sqrt{1} \) is 1 and \( \sqrt{4} \) is 2.

Fraction Simplification

Simplifying fractions is the process of reducing the fraction to its simplest form. This involves finding a number that divides into both the numerator and the denominator. This number is known as the greatest common divisor (GCD).

In the context of the original problem, we start with \( \frac{9}{36} \). The GCD of 9 and 36 is 9. So, you can simplify the fraction by dividing both the numerator and the denominator by 9. This results in the simplified fraction \( \frac{1}{4} \).

Simplifying fractions makes it easier to perform operations like finding square roots and ensures your final answer is in its simplest form. Whenever you encounter a fraction, always check to see if it can be simplified by identifying and dividing by their GCD.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operations. They help in representing generalized patterns and solutions in mathematics. In the given exercise, you have an algebraic expression within the square root: \( \sqrt{\frac{9}{36} x^4} \).

Variables like \( x \) are placeholders that can represent numbers. When you see an expression like \( x^4 \), it means \( x \) is raised to the power of 4, which is \( x \times x \times x \times x \). Simplifying algebraic expressions follows the rules of arithmetic while paying special attention to variables and their exponents.

For the exercise, notaably: you have \( x^4 \) inside the square root. By applying the property of square roots and exponents \( \sqrt{x^n} = x^{n/2} \), you find that \( \sqrt{x^4} \) simplifies to \( x^2 \). The final expression becomes \( \frac{1}{2} x^2 \), simplifying the algebraic expression to its neatest form.