Problem 10 - Intermediate Algebra

Question

Add or subtract. $$ \sqrt{4 x^{7}}+9 x^{2} \sqrt{x^{3}}-5 x \sqrt{x^{5}} $$

Step-by-Step Solution

Answer

The expression simplifies to \( 6x^3 \sqrt{x} \).

Step 1: Simplify Each Term

Start by simplifying each term with the square root. The expression is \( \sqrt{4x^7} + 9x^2 \sqrt{x^3} - 5x \sqrt{x^5} \). - First term: \( \sqrt{4x^7} = \sqrt{4} \cdot \sqrt{x^7} = 2x^{3.5} \). Since it should be expressed with integer exponents, it becomes \( 2x^3 \sqrt{x} \).- Second term: \( 9x^2 \sqrt{x^3} = 9x^2 \cdot x^{1.5} = 9x^{3.5} \). It simplifies to \( 9x^3 \sqrt{x} \).- Third term: \( -5x \sqrt{x^5} = -5x \cdot x^{2.5} = -5x^{3.5} \). It simplifies to \( -5x^3 \sqrt{x} \).

Step 2: Combine Like Terms

Now that all terms are in a similar form, combine them. All terms are like terms because they include \( x^3 \sqrt{x} \).- Combine the coefficients: \( 2x^3 \sqrt{x} + 9x^3 \sqrt{x} - 5x^3 \sqrt{x} \) simplifies to \( (2 + 9 - 5)x^3 \sqrt{x} \).- The simplified result is \( 6x^3 \sqrt{x} \).

Step 3: Conclusion: Present the Result

The simplified expression after combining like terms is \( 6x^3 \sqrt{x} \).

Key Concepts

Combining Like Terms

Combining like terms is an essential algebraic process that simplifies expressions. When dealing with algebraic expressions, like terms are the terms that have identical variable parts raised to the same power. In simple terms, these are the terms where you can just add or subtract the coefficients as the variables and their exponents are the same.

In the original exercise, the expression is made up of terms like \(2x^3 \sqrt{x}\), and all the terms contain this repeated element.
When you encounter like terms:

• Identify terms that have the same variable factors and exponents.
• Add or subtract the coefficients of these terms while keeping the variable and exponent part unchanged.

During this exercise, we simplified it from \(2x^3 \sqrt{x} + 9x^3 \sqrt{x} - 5x^3 \sqrt{x}\) to \(6x^3 \sqrt{x}\) by simply adding and subtracting the coefficients \(2\), \(9\), and \(-5\), because they are all like terms.The process clears up the expression to something shorter and more manageable, which is handy for further mathematical operations.

Integer Exponents

Working with integer exponents is crucial in simplifying expressions involving indices. An exponent tells you how many times to multiply the base by itself. When simplifying expressions, we often aim to work with integer exponents because they provide a neater format compared to fractional exponents.

In the given exercise, terms initially appear with fractional exponents during the square root simplification process, such as \(x^{3.5}\). But expressing these as integer exponents with radicals in the simplest form, like \(x^3 \sqrt{x}\), makes computations more straightforward.

To convert fractional exponents to integer exponents:

• Identify the part of the exponent that can be expressed as an integer.
• Write the remaining fractional part as a separate radical factor.

This method keeps expressions neat and ensures they follow conventional mathematical expressions, as shown when simplifying \(2x^{3.5}\) into \(2x^3 \sqrt{x}\).
Understanding and handling integer exponents proficiently is a foundational skill that underpins more complex algebraic manipulations.

Square Root Simplification

Simplifying square roots is about expressing them in their most manageable and straightforward form. Square root simplification often involves breaking down a complicated radical expression into simpler components by factoring out perfect squares and simplifying them.For example, in the first step of simplification, the term \(\sqrt{4x^7}\) can be broken down:

- Separate the square root of the constant \(\sqrt{4} = 2\).- Factor the variable component as \(x^7\) into \(x^6 \times x^1\), turning into \(x^3 \sqrt{x}\).The goal is to eliminate any squares under the root and show the expression clearly. This technique is applied similarly to all terms so they can be combined more easily later based on like terms, as in the expression \(\sqrt{x^5} = x^2 \sqrt{x}\).When handling square roots:

• Break down the number or variable under the root into known squares.
• Simplify each part independently.

Mastering square root simplification allows one to rewrite and solve expressions significantly more efficiently, paving the way for further algebraic manipulations.