Problem 30 - College Algebra with Modeling and Visualization

Question

Simplify the expression. Assume that all variables are positive. $$ \sqrt{16 x^{4} y^{6}} $$

Step-by-Step Solution

Answer

\(4x^2y^3\)

Step 1: Apply the relevant algebraic rules

Use properties of exponents, radicals, or algebraic identities to simplify the expression.

Step 2: State the simplified result

The simplified expression is \(4x^2y^3\).

Key Concepts

Square Roots

Square roots are a fundamental concept in mathematics, especially when simplifying expressions. The square root of a number is a value that, when multiplied by itself, gives the original number. For example, the square root of 16 is 4, because \(4 \times 4 = 16\).

When dealing with square roots in algebraic expressions, the properties of square roots can greatly simplify calculations. One important property is that the square root of a product is equal to the product of the square roots. This property allows us to separate the terms under the square root sign into individual parts that can be more easily simplified.

In our exercise, we have \(\sqrt{16x^4y^6}\). Using the property of square roots, we separate it as \(\sqrt{16} \cdot \sqrt{x^4} \cdot \sqrt{y^6}\). Breaking down the square root of a number into simpler components makes the simplification process more manageable.

Exponents

Exponents are used in mathematics to denote repeated multiplication of a number by itself. In an expression like \(x^4\), the base \(x\) is raised to the power of 4, meaning \(x\) is multiplied by itself four times: \(x \cdot x \cdot x \cdot x\).

When simplifying expressions involving exponents, one crucial rule is that the square root of an expression with an even exponent can be easily simplified. For example, the square root of the expression \(x^4\) is \(x^{4/2} = x^2\).

In our specific problem, we apply this rule to simplify \(\sqrt{x^4}\) to \(x^2\). The same logic applies to \(y^6\), where \(\sqrt{y^6} = y^{6/2} = y^3\). Understanding exponent rules allows for a smooth simplification process of more complex algebraic expressions.

Algebraic Expressions

Algebraic expressions consist of numbers, variables, and exponents that represent mathematical relationships. Simplifying these expressions involves writing them in their simplest form while preserving their value.

In the context of our exercise, the algebraic expression \(16x^4y^6\) is under a square root. By applying the properties of square roots and exponents, we can simplify this expression effectively.

First, recognize each component: the numeric part \(16\) becomes \(4\) when simplified under the square root, since \(\sqrt{16} = 4\). Then, simplify each variable component using exponent rules: \(x^4\) becomes \(x^2\) and \(y^6\) simplifies to \(y^3\).

• The initial expression: \(\sqrt{16x^4y^6}\)
• Simplification step-by-step: \(\sqrt{16} \cdot \sqrt{x^4} \cdot \sqrt{y^6} = 4 \cdot x^2 \cdot y^3 = 4x^2y^3\)

Through these simplifications, the original expression is reduced to its simplest form, which makes it easier to interpret and work with further algebraic operations.