Problem 74 - Elementary Algebra

Question

For the following problems, simplify each of the radical expressions. $$ \sqrt{\frac{27 x^{6} y^{15}}{3^{3} x^{3} y^{5}}} $$

Step-by-Step Solution

Answer

Question: Simplify the following radical expression: $\sqrt{\frac{27x^6y^{15}}{3^3x^3y^5}}$. Answer: The simplified radical expression is $x^{\frac{3}{2}}y^5$.

Step 1: Break down the terms inside the square root

First, let's break down the expression inside the square root: $$ \sqrt{\frac{27 x^{6} y^{15}}{3^{3} x^{3} y^{5}}} $$ To break down the numerator and denominator, we can write 27 as 3^3, and divide the exponents of x and y by 3: $$ \sqrt{\frac{3^3 x^{6} y^{15}}{3^3 x^{3} y^{5}}} $$

Step 2: Cancel terms inside the square root

Now, we can cancel out the similar terms in the numerator (top) and denominator (bottom) under the square root symbol: $$ \sqrt{\frac{3^3 x^{6} y^{15}}{3^3 x^{3} y^{5}}} = \sqrt{\frac{x^{6} y^{15}}{x^{3} y^{5}}} $$

Step 3: Apply the rule for dividing exponents

When dividing two terms with the same base and exponents, we subtract the exponents: $$ \sqrt{\frac{x^{6} y^{15}}{x^{3} y^{5}}} = \sqrt{\frac{x^{(6-3)}y^{(15-5)}}{1}} = \sqrt{x^{3}y^{10}} $$

Step 4: Simplify the square root

Now that we have a simplified expression inside the square root, we can use the rule for square roots and exponents: $$ \sqrt{x^ay^b} = x^{\frac{a}{2}}y^{\frac{b}{2}} $$ Applying this rule to our expression, we get: $$ \sqrt{x^{3}y^{10}} = x^{\frac{3}{2}}y^{\frac{10}{2}} = x^{\frac{3}{2}}y^{5} $$ The simplified radical expression is: $$ x^{\frac{3}{2}}y^5 $$

Key Concepts

Square Root Simplification

When simplifying square roots, the goal is to make the radical expression as simple as possible. A radical can often be simplified by breaking it down into prime factors and identifying perfect squares. Simplification involves finding factors of the number under the radical that are square numbers and then taking them out from under the radical, essentially 'reducing' the expression. For example, with \( \sqrt{27} \), we would observe that 27 is 3 raised to the 3rd power, or \( 3^3 \). Since \( 3^2 \), or 9, is a perfect square, we can take the square root of 9 out from under the radical, leaving us with \( 3 \sqrt{3} \).

This process can also apply to variables, where the exponent indicates how many times the variable is multiplied by itself. If the exponent is even, it means that there are pairs of the variable that can be taken out of the square root. For instance, \( x^6 \) can be simplified to \( x^3 \) outside of the square root because 6 is divisible by 2, signifying three pairs of \( x \) being multiplied together.

Exponent Rules

Exponent rules, or laws of exponents, are fundamental in dealing with radicals in algebra. They provide a framework for simplifying expressions that involve powers. The most relevant rules for radical simplification are:

• Product of Powers: When multiplying two powers with the same base, add the exponents (e.g., \( x^a \cdot x^b = x^{a+b} \)).
• Quotient of Powers: When dividing two powers with the same base, subtract the smaller exponent from the larger one (e.g., \( \frac{x^a}{x^b} = x^{a-b} \), if \( a > b \)).
• Power of a Power: When raising a power to another power, multiply the exponents (e.g., \( (x^a)^b = x^{ab} \)).
• Power of a Product: To raise a product to a power, raise each factor to the power (e.g., \( (xy)^a = x^a y^a \)).

Correct application of these rules is essential when working with radical expressions, as they provide the basis for simplification and manipulation of algebraic terms.

Radicals in Algebra

Radicals in algebra often involve the square root symbol, and they can be intimidating at first glance. However, they follow specific arithmetic rules that allow for simplification and operations like addition, subtraction, multiplication, and division. A radical can contain not just numbers but also variables and even expressions.

One of the key steps in simplifying a radical is to eliminate any radicals in the denominator, a process known as 'rationalizing the denominator.' This is done by multiplying the numerator and denominator by a number or expression that will get rid of the radical in the denominator. Another important aspect is to simplify the expression inside the radical as much as possible before performing any operations, which may involve factoring or using the exponent rules to combine and simplify like terms.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can contain numbers, variables, and operations (like addition, subtraction, multiplication, and division). Simplifying algebraic expressions is a process of combining like terms and reducing the expression to the fewest terms possible.

An important part of working with algebraic expressions is understanding how to handle exponents and radicals since they frequently appear in algebra. Knowing how to rearrange and combine terms using the distributive, commutative, and associative properties can greatly simplify complex expressions. The ultimate goal with any algebraic expression is to make it readable and as uncomplicated as possible, thereby making it easier to solve equations and understand the relationships between variables.