Problem 55 - Intermediate Algebra

Question

Simplify. See Examples 3 and 4 $$ \sqrt{9 x^{7} y^{9}} $$

Step-by-Step Solution

Answer

The simplified form is \(3x^3 y^4 \sqrt{xy}\).

Step 1: Break Down the Expression

The expression \( \sqrt{9 x^7 y^9} \) involves a square root acting on several factors. We can simplify it by addressing each part separately: the constant \(9\), the variable \(x^7\), and the variable \(y^9\).

Step 2: Simplify the Constant

Since \(9\) is a perfect square, we find \( \sqrt{9} = 3 \). This simplifies the constant part of our expression.

Step 3: Simplify the Exponent of x

The exponent of \(x\) is 7, which is odd. We can separate it into even and odd parts: \(x^7 = x^6 \, x^1 = (x^3)^2 \, x^1\). Thus, \( \sqrt{x^7} = x^3 \sqrt{x}\).

Step 4: Simplify the Exponent of y

The exponent of \(y\) is 9, which is odd. We divide it into \(y^8 \, y^1 = (y^4)^2 \, y^1\). Thus, \( \sqrt{y^9} = y^4 \sqrt{y}\).

Step 5: Combine the Simplified Parts

Combine the results from the previous steps: \( 3 \cdot x^3 \sqrt{x} \cdot y^4 \sqrt{y} = 3x^3 y^4 \sqrt{xy} \). This is the simplified form of the original expression.

Key Concepts

Square Roots

Square roots are a fundamental concept in mathematics that give us any number that, when multiplied by itself, produces the original number under the radical. For example, the square root of 9 is 3 because when 3 is multiplied by itself, it gives us 9.
When we see the square root symbol, written as \( \sqrt{} \), it indicates the operation of finding this root. In the exercise, we were tasked with simplifying \( \sqrt{9x^{7}y^{9}} \), which involved finding square roots of both a constant and variables raised to exponents.

• Identify perfect squares: The number inside the square root that can be written as a power of 2 is a perfect square—for instance, 9 is \( 3^2 \).
• Use factored expressions: For variables with exponents inside the square root, factor their powers into even components, since \( x^2 \) makes a perfect square.
• After simplifying these perfect squares, extract them outside of the square root as their base number or variables raised to the power of half the original exponent.

Thus, understanding square roots involves recognizing perfect squares and simplifying expressions by rearranging and breaking down complex numbers and variables.

Algebraic Expressions

Algebraic expressions allow us to perform mathematical operations involving numbers, variables, and their exponents in a convenient and symbolic form. These expressions can become quite complex but follow rules that simplify solving or evaluating them.
In the case of \( \sqrt{9x^{7}y^{9}} \), the expression includes a numerical part and variables raised to different powers. Handling each part separately is key to simplifying such expressions.

• Navigating constants: Recognize numbers like 9 in the expression as regular constants that might simplify neatly when they are perfect squares.
• Managing variables: Observe variables within the expressions with an eye on their exponents. This is essential for separating even and odd powers to simplify accurately.
• Combining results: Once each part of the algebraic expression is simplified, you combine them for the final simplified result.

By understanding and applying these techniques, algebraic expressions become much easier to work with, allowing for simplification even when the original expression involves square roots or significant exponents.

Exponent Rules

Exponent rules provide the framework necessary to handle the powers of numbers and variables, streamlining processes like simplifying expressions. These rules cover how exponents interact through multiplication, division, and toowers, which is crucial when working with square roots and algebraic expressions.
In the given exercise, certain exponent strategies helped in simplifying \( \sqrt{9x^{7}y^{9}} \):

• Power breakdown: Decompose any exponent into parts that can be simplified separately, such as turning odd exponents into a combination of even and odd components.
• Rewriting exponents: Convert powers into products, as in taking \( x^{7} \) to \( (x^{3})^{2} \times x \), enabling the simplification through square roots.
• Odd versus even: Identify whether exponents are odd or even to determine how they can be split down for easier handling.
• Root extraction: Remember that when something is squared, its square root returns it to its base, such as using \( (x^{3})^{2} \) to find \( x^{3} \).

Utilizing exponent rules makes it feasible to break down even complex expressions into manageable parts for calculation or further simplification.