Problem 55 - Algebra A Combined Function

Question

Simplify. Assume that all variables represent positive real numbers. $$ \sqrt{9 x^{7} y^{9}} $$

Step-by-Step Solution

Answer

\( 3x^3y^4 \sqrt{xy} \)

Step 1: Break down the expression

To simplify \( \sqrt{9 x^{7} y^{9}} \), you can break it down into separate square roots: \( \sqrt{9} \cdot \sqrt{x^{7}} \cdot \sqrt{y^{9}} \). This makes it easier to simplify each part individually.

Step 2: Simplify \( \sqrt{9} \)

Calculate the square root of 9. Since \( 9 = 3^2 \), we have \( \sqrt{9} = 3 \). This is straightforward as it involves recognizing the square number.

Step 3: Simplify \( \sqrt{x^{7}} \)

The exponent on \( x \) is 7. Divide the exponent by 2 to find the whole numbers and remainder: \( x^{7} = x^{6 + 1} = x^6 \cdot x^1 \). Now, \( \sqrt{x^7} = \sqrt{x^6} \cdot \sqrt{x^1} = x^3 \cdot \sqrt{x} \), since \( x^6 = (x^3)^2 \).

Step 4: Simplify \( \sqrt{y^{9}} \)

The exponent on \( y \) is 9. Similarly, divide the exponent by 2: \( y^{9} = y^{8 + 1} = y^8 \cdot y^1 \). Thus, \( \sqrt{y^9} = \sqrt{y^8} \cdot \sqrt{y^1} = y^4 \cdot \sqrt{y} \), because \( y^8 = (y^4)^2 \).

Step 5: Combine Simplifications

Put it all together: \( 3 \cdot x^3 \cdot \sqrt{x} \cdot y^4 \cdot \sqrt{y} = 3x^3y^4 \cdot \sqrt{xy} \). This combines the simplified square roots with the remaining roots.

Key Concepts

Exponent Rules

When simplifying radical expressions, one often uses exponent rules to manage the base numbers raised to certain powers. Exponent rules help in breaking down complex expressions into more manageable parts and determine how many times a number, known as the base, is multiplied by itself.

• Basic Rules: The primary rule is that multiplying powers with the same base, you add the exponents: \( a^m \times a^n = a^{m+n} \).
• Power of a Power: Raising a power to another power involves multiplying the exponents: \( (a^m)^n = a^{m \times n} \).
• Power of a Product: This applies when a product is raised to an exponent: \( (ab)^n = a^n b^n \).

With our example, \( x^7 \) becomes \( x^{6+1} \), which you then separate using these rules to simplify to \( x^3 \cdot \sqrt{x} \). Similarly, \( y^9 \) can be expressed as \( y^{8+1} \) and simplified to \( y^4 \cdot \sqrt{y} \). Understanding these rules allows for methodical simplification of expressions into more elementary forms.

Simplification Techniques

Simplification techniques aim to express mathematical expressions in the most reduced form possible, removing unnecessary complexities. Simplifying allows easier computations and provides clear insights into the properties of the expression.

• Breaking Down Components: Start by identifying and separating parts of the expression. For example, \( \sqrt{9x^7y^9} \) is split as \( \sqrt{9} \cdot \sqrt{x^7} \cdot \sqrt{y^9} \).
• Geometric Interpretation: Recognize perfect squares or cubes that make solving much easier. Since \( 9 = 3^2 \), simplifying becomes straightforward because \( \sqrt{9} = 3 \).

The goal is to reduce the expression to its simplest form, eliminating extraneous elements. Applying simplification involves recognizing patterns or special forms, such as exponent rules and square roots, which can make the problem-solving process more intuitive.

Algebraic Expressions

Algebraic expressions consist of variables, constants, and operations that connect them, forming parts of mathematical statements or equations. They serve as the foundational language of mathematics, allowing for representation and manipulation of quantities.

• Variables and Constants: Variables, like \( x \) and \( y \), represent unknown values, while constants such as numbers (e.g., 9) have fixed values.
• Operations: These include addition, subtraction, multiplication, division, and exponentiation, combining variables and constants in formulas and equations.
• Expression Forms: Expressions might be polynomial (e.g., \( ax^n + bx^{n-1} + \ldots + c \)), rational, or radical, like in our exercise, where expressions under a square root need to be managed.

In the exercise, dealing with \( \sqrt{9x^7y^9} \) involves understanding these components and simplifying them through algebraic manipulations. Recognizing these forms and how to transform or simplify them is key to solving algebraic problems efficiently.