Problem 19 - Algebra 1

Question

Simplify the expression. $$\frac{1}{3} \sqrt{54}$$

Step-by-Step Solution

Answer

The simplified expression is \(\sqrt{6}\)

Step 1: Prime Factorization of 54

The first step is to break down 54 into its prime factors. Prime factorization of 54 is \(2 \times 3 \times 3 \times 3\). So we can write square root of 54 as \(\sqrt{2 \times 3 \times 3 \times 3}\)

Step 2: Simplify the Square Root

We also know that the square root of \(a \times b\) is equivalent to \(\sqrt{a} \times \sqrt{b}\). Using this, \(\sqrt{2 \times 3 \times 3 \times 3} = \sqrt{2} \times \sqrt{3 \times 3} \times \sqrt{3}\). This can be simplified further to \(\sqrt{2} \times 3 \times \sqrt{3} = 3\sqrt{2} \sqrt{3}\)

Step 3: Divide the Simplified Square Root by the Fraction

Finally, we have to divide \(3\sqrt{2} \cdot \sqrt{3}\) by \(\frac{1}{3}\). This would result in \(3\cdot \frac{1}{3}\sqrt{2} \cdot \sqrt{3} = \sqrt{2} \cdot \sqrt{3} = \sqrt{6}\)

Key Concepts

Prime Factorization

When faced with simplifying radical expressions, prime factorization is an essential initial step. It involves breaking down a composite number into the product of its prime factors, which are the building blocks of all numbers. These factors are prime numbers, which means they are greater than 1 and can only be divided by 1 and themselves without leaving a remainder.

To carry out prime factorization, we start by finding the smallest prime number that divides our given number and then continue the process with the quotient until we reach 1. For example, with the number 54, we begin by dividing it by 2, the smallest prime number, to get 27. We then divide 27 by 3 repeatedly until we are left with 1. The prime factors of 54 are therefore 2 and 3, with 3 being repeated three times, expressed as \(2 \times 3 \times 3 \times 3\).

Understanding prime factorization is crucial as it lays the foundation for simplifying square roots and handling complex algebraic expressions.

Radical Expressions

Radical expressions involve numbers under the radical sign, such as square roots, cube roots, and other higher-order roots. The square root of a number is a value that, when multiplied by itself, gives the original number. For instance, the square root of 25 is 5, since \(5 \times 5 = 25\).

In simplifying radical expressions, it's essential to identify and extract perfect squares from under the radical sign. A perfect square is a number that has a whole number as its square root, like 16, which is \(4 \times 4\).

We can simplify the square root of a number by breaking it apart using the properties of square roots: \(\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}\). Once in simpler form, the expression can be manipulated using the given rules and operations, leading to simplified expressions that may reveal more perfect squares or become part of a more complex algebraic equation.

Square Root Properties

Square roots abide by certain properties that make simplifying radical expressions easier. A key property is that the square root of a product is equal to the product of the square roots: \(\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}\). This property is pivotal in breaking down composite numbers within a square root into more manageable factors.

Another vital property to remember is that the square root of a quotient is the quotient of the square roots, which means that \(\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}\). Also, when dealing with fractions that contain square roots, the fraction itself can multiply or divide the values under the square root, simplifying the expression even further.

These properties are not just academic; they're practical tools that enable students to tackle the square roots of large or complex numbers with confidence. By internalizing these rules, one can simplify square roots efficiently and effectively.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can contain numbers, variables, and operators (such as add, subtract, multiply, and divide). They don't have an equals sign, as equations do, but they're a crucial part of forming equations and solving algebraic problems.

When working with algebraic expressions, we need to apply the order of operations appropriately: first simplifying expressions within parentheses, then exponents, followed by multiplication and division from left to right, and finally addition and subtraction from left to right.

Incorporating prime factorization and the properties of square roots helps in managing even the most unruly algebraic expressions. By simplifying square roots within these expressions, one clarifies the path to solving for variables or further manipulating the equation. On going practice with radical expressions and square root properties adeptly prepares students for the myriad challenges presented within algebra.