Problem 10 - Algebra A Combined Function

Question

Add or subtract as indicated. Assume that all variables represent positive real numbers. $$ \sqrt{4 x^{7}}+9 x^{2} \sqrt{x^{3}}-5 x \sqrt{x^{5}} $$

Step-by-Step Solution

Answer

The simplified expression is \(6x^{7/2}\).

Step 1: Simplify Each Radical Expression

Start by simplifying each of the radical expressions in the sum. The expression is \( \sqrt{4x^7} + 9x^2 \sqrt{x^3} - 5x \sqrt{x^5} \).- For \( \sqrt{4x^7} \), recognize that this can be written as \( \sqrt{4} \cdot \sqrt{x^6} \cdot \sqrt{x} = 2x^3 \sqrt{x} \).- For \( 9x^2 \sqrt{x^3} \), simplify \( \sqrt{x^3} \) to \( x^{3/2} \), resulting in \( 9x^2 \cdot x^{3/2} = 9x^{2+3/2} = 9x^{7/2} \).- For \( -5x \sqrt{x^5} \), simplify \( \sqrt{x^5} \) to \( x^{5/2} \), resulting in \( -5x \cdot x^{5/2} = -5x^{1+5/2} = -5x^{7/2} \).

Step 2: Simplify and Combine Like Terms

Combine terms that have the same radical or exponential form:- Our simplified expression from Step 1 is \( 2x^3\sqrt{x} + 9x^{7/2} - 5x^{7/2} \).- Combine the like terms \( 9x^{7/2} - 5x^{7/2} \) to get \( 4x^{7/2} \).Thus, the expression simplifies to \( 2x^3\sqrt{x} + 4x^{7/2} \).

Step 3: Express the Simplified Terms Uniformly

Both terms contain powers of \( x \), try to express them in a consistent way where possible:- Both terms involve fractions as exponents or radicals.- Recognizing \( 2x^3 \sqrt{x} \) as \( 2x^{3+1/2} = 2x^{7/2} \).- Now the combined expression can be expressed uniformly as \( 2x^{7/2} + 4x^{7/2} \).Combine these like terms to form the final simplified expression.

Step 4: Final Combination

Add the coefficients of the like terms:- We now combine \( 2x^{7/2} + 4x^{7/2} \) which results in \( (2+4)x^{7/2} = 6x^{7/2} \).This is the final simplified expression.

Key Concepts

Real Numbers

Real numbers are fundamental in mathematics and include all the numbers you can think of on a number line. They encompass both rational and irrational numbers, allowing you to perform operations like addition, subtraction, multiplication, and division. In the given exercise, we assume all variables represent positive real numbers. This assumption simplifies calculations and ensures meaningful interpretations.

• Rational numbers are those that can be written as fractions, like integers and simple fractions.
• Irrational numbers are those that cannot be precisely expressed as fractions — for example, the square root of a non-perfect square or pi (\(\pi\)).

Understanding real numbers is essential because it helps us to build upon various mathematical concepts, such as the simplification of radical expressions, which we will explore in detail next.

Radicals Simplification

Simplifying radicals is crucial for making complex expressions easier to work with. A radical expression, like a square root, refers to numbers that can be expressed in root form. When simplifying, the goal is to break down these expressions into their simplest forms.

Steps in Simplification

The initial exercise involves simplifying multiple radical expressions by:

• Breaking down the components inside the radical into perfect squares or cubes.
• Using the property \( \sqrt{a \times b} = \sqrt{a} \times \sqrt{b} \).

For example, \( \sqrt{4x^7} \) can be expressed as \( \sqrt{4} \times \sqrt{x^6} \times \sqrt{x} \), simplifying to \( 2x^3 \sqrt{x} \). This technique helps in expressing terms like radicals uniformly, allowing easy arithmetic operations such as addition and subtraction. Simplification also enables better visual understanding of the structure and behavior of expressions.

Exponent Rules

Exponent rules are the foundation for simplifying expressions in mathematics involving powers. They allow us to manipulate terms and solve problems more efficiently. Different exponent rules can be applied to simplify and solve expressions like those in our exercise.

Important Rules

• Product of Powers: \( x^a \times x^b = x^{a+b} \) - This is used to combine terms with the same base.
• Power of a Power: \( (x^a)^b = x^{ab} \) - Useful when handling nested exponential terms.
• Simplifying Radicals to Exponents: For instance, \( \sqrt{x} = x^{1/2} \) simplifies expression handling.

In our specific exercise, exponent rules were applied to simplify terms like \( 9x^2 \sqrt{x^3} \) to \( 9x^{7/2} \) through operations like converting radicals to fractional exponents. This uniform approach using exponent rules provides efficiency when dealing with expression simplification and combination.