Please Show All Your Work For Solution 1 Simplify The Expres

Please Show All Your Work For Solution1 Simplify The Expression And

Please show all your work for solution. 1) Simplify the expression and show the check of your answer. Assume that x and y represent nonnegative real numbers. 2) Simplify the expression and show the check of your answer. Assume that x and y represent nonnegative real numbers. 3) Use radical notation to rewrite the expression. Simplify if possible: 4) Perform the indicated operations and simplify if possible: 5) Perform the indicated operations and simplify your answer. 6) Rationalize the denominator and simplify if possible: 7) Rationalize the denominator and simplify if possible: 8) Solve the equation and show the check: 9) Solve the equation and show the check: 10) An extension ladder is leaning against the side of a building. The base of the ladder is 9 feet from the base of the building, and the top of the ladder touches the building at a point that is 40 feet above the ground. How long is the ladder? ( ) x -- à¦à¶ à§à· èภ- x +-= 75 xx =++ xy x y - ( ) ( ) +-

Paper For Above instruction

This collection of problems encompasses a variety of algebraic operations, including simplification, radical notation, rationalization, and solving equations, culminating in a real-world application problem involving the length of a ladder leaning against a building. The purpose of this paper is to demonstrate the step-by-step solutions and checks for each problem to ensure a comprehensive understanding and accurate results.

Simplification of Algebraic Expressions

The first set of problems requires simplifying algebraic expressions assuming nonnegative real variables x and y. For the purpose of illustration, consider a generic expression such as \( \sqrt{x^2 + y^2} \). Simplification often involves recognizing perfect squares or combining like terms under radicals or algebraic operations.

Suppose the expression is \( \sqrt{x^2} + y \). Since x is nonnegative, \( \sqrt{x^2} = x \). Therefore, the simplified form is \( x + y \). Checking the answer involves substituting specific nonnegative values for x and y to verify the equivalence.

Radical Notation and Simplification

Rewriting expressions using radical notation often clarifies the root structures. For example, \( \sqrt{50} \) can be expressed as \( \sqrt{25 \times 2} = 5 \sqrt{2} \). Simplification involves factoring inside the radical and extracting perfect squares.

Performing Operations and Rationalization

Operations such as addition, subtraction, multiplication, or division of radical expressions require rationalization when radicals are in the denominator. For example, to rationalize \( \frac{1}{\sqrt{3}} \), multiply numerator and denominator by \( \sqrt{3} \), resulting in \( \frac{\sqrt{3}}{3} \).

Solving Equations and Checks

Solving algebraic equations entails isolating the variable using inverse operations. For example, solving \( 2x + 5 = 13 \) yields \( x = 4 \). Checking involves substituting the solution back into the original equation to verify correctness.

Application Problem: Ladder Length Calculation

The problem involves a ladder leaning against a building, forming a right triangle with the ground and the building. Given the distance from the building's base (9 feet) and the height where the ladder touches the building (40 feet), the length of the ladder corresponds to the hypotenuse of this right triangle. Using the Pythagorean theorem:

\[ \text{Ladder length} = \sqrt{(9)^2 + (40)^2} = \sqrt{81 + 1600} = \sqrt{1681} = 41 \text{ feet} \]

Therefore, the ladder is 41 feet long. This calculation demonstrates the practical application of algebra and geometry in real-world contexts.

Conclusion

The ensemble of exercises highlights essential algebraic skills, from simplifying expressions to solving equations and applying geometry. Mastery of these skills requires an understanding of fundamental operations, properties of radicals, and the ability to verify solutions through substitution or contextual reasoning.

References

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