Problem 22 - Elementary Algebra

Question

For the following problems, simplify the expressions. $$ \sqrt{x}\left(\sqrt{x^{5}}-\sqrt{3 x}\right) $$

Step-by-Step Solution

Answer

Question: Simplify the expression $$\sqrt{x}(x^{5\sqrt{x}}-3x\sqrt{x})$$. Answer: The simplified expression is $$x^{3} - \sqrt{3}x$$.

Step 1: Distributing the terms inside the parenthesis

Distribute the $$\sqrt{x}$$ to both terms inside the parenthesis: $$ \sqrt{x}\times\sqrt{x^{5}} - \sqrt{x}\times\sqrt{3x} $$

Step 2: Simplify the radicals using exponent rules

Use the rule $$\sqrt{a} \times \sqrt{b} = \sqrt{ab}$$ and simplify the radicals: $$ \sqrt{x \cdot x^{5}} - \sqrt{x \cdot 3x} $$

Step 3: Simplify the expressions inside the radicals

Combine the terms and simplify the expressions inside the radicals: $$ \sqrt{x^{6}} - \sqrt{3x^{2}} $$

Step 4: Write the expressions with a rational exponent

Replace the square roots with rational exponents: $$ (x^{6})^{\frac{1}{2}} - (3x^{2})^{\frac{1}{2}} $$

Step 5: Apply exponent rules to simplify the expressions

Apply the exponent rule $$(a^{m})^n = a^{mn}$$ to simplify the expressions: $$ x^{(6\cdot\frac{1}{2})} - 3^{\frac{1}{2}}\cdot x^{(2\cdot\frac{1}{2})} $$

Step 6: Calculate and write the final simplified expression

Calculate the exponents and write the simplified expression: $$ x^{3} - \sqrt{3}x $$ The simplified expression is $$x^{3} - \sqrt{3}x$$.

Key Concepts

Radicals and Rational Exponents

Understanding the relationship between radicals and rational exponents is crucial for simplifying algebraic expressions that contain square roots, cube roots, and so on. A radical can be expressed as a rational exponent, which is another way of writing roots. For example, the square root of a number, such as \( \sqrt{x} \), can be written with a rational exponent as \( x^{1/2} \). Similarly, the cube root, \( \sqrt[3]{x} \), is equivalent to \( x^{1/3} \).

This conversion is based on the general rule \( \sqrt[n]{a} = a^{1/n} \), where \( n \) is the index of the radical (which is 2 for square roots, 3 for cube roots, etc.), and \( a \) is the radicand, the number under the radical sign. Once the expression is in exponential form, it can be easier to manipulate using exponent rules, leading to a simplified form.

When simplifying algebraic expressions with radicals during the exercise, as in the step from \( \sqrt{x^{6}} \) to \( x^{3} \), we applied this concept. We expressed the square root as a rational exponent \( x^{6\frac{1}{2}} \) and then simplified to \( x^{3}\), since the exponent was halved due to the square root.

Exponent Rules

Exponent rules, also known as laws of exponents, are critical tools for simplifying expressions involving exponents. There are several key rules to remember:

• The Product Rule: \( a^m \cdot a^n = a^{m+n} \), which states that when multiplying two exponents with the same base, you add the exponents.
• The Quotient Rule: \( a^m / a^n = a^{m-n} \) indicates that when dividing two exponents with the same base, you subtract the exponents.
• The Power Rule: \( (a^m)^n = a^{mn} \), which tells us that when raising an exponent to another power, you multiply the exponents.

In the original exercise, we used the Power Rule to simplify \( (x^{6})^{\frac{1}{2}} \) to \( x^{3} \) because we were multiplying the exponents \(6\) and \(\frac{1}{2}\).

It's also essential to remember that any number raised to the zero power is equal to one, and that negative exponents indicate the reciprocal of the base raised to the positive exponent. These rules are fundamental in algebra and are used to simplify complex expressions efficiently.

Algebraic Simplification

Algebraic simplification is the process of reducing expressions to their simplest form. This involves a number of skills, including factoring, combining like terms, and applying the aforementioned exponent rules. The goal is to make the expression as concise and straightforward as possible, without changing its value. Simplification may include eliminating parentheses, combining like terms, and reducing fractions.

In the context of our exercise, algebraic simplification occurred in several steps. For instance, initially we distributed \( \sqrt{x} \) across the terms in the parentheses. Later, we simplified the radicals and combined terms. During these steps, we're looking out for any opportunity to make the expression cleaner or more compact.

The final simplified expression, \( x^{3} - \sqrt{3}x \) that we achieved is much easier to understand and work with than the original. It is important to practice these simplification techniques regularly, as they are widely used in various fields of mathematics and applied sciences.