Equivalent Expressions For 5x√(28x⁵) + 8x³√(7x) Simplification Guide

In the realm of mathematics, simplifying expressions involving radicals is a fundamental skill. Radicals, often appearing complex at first glance, can be elegantly transformed into simpler forms by applying the rules of algebra and a keen understanding of their properties. In this comprehensive exploration, we will delve into the intricate process of identifying expressions equivalent to $5x[sqrt{28x^5} + 8x^3[sqrt{7x}$, where $x > 0$. This exercise not only tests our ability to manipulate radicals but also reinforces the importance of recognizing equivalent forms in algebraic expressions. Let's embark on this mathematical journey, unraveling the complexities and discovering the beauty of simplification.

Understanding the Basics of Radical Simplification

Before we dive into the specifics of the given expression, it's crucial to solidify our understanding of radical simplification. Radicals, represented by the symbol $[sqrt{}$, indicate the root of a number. For instance, $[sqrt{9}$ represents the square root of 9, which is 3. Simplifying radicals involves expressing them in their simplest form, where the radicand (the number under the radical) has no perfect square factors (for square roots), perfect cube factors (for cube roots), and so on.

The key principles guiding radical simplification are:

1. Product Property of Radicals: $[sqrt{ab} = [sqrt{a} \cdot [sqrt{b}$
2. Quotient Property of Radicals: $[sqrt{\frac{a}{b}} = \frac{[sqrt{a}}{[sqrt{b}}$
3. Simplifying Radicals with Variables: When dealing with variables under radicals, we look for exponents that are multiples of the index (the small number indicating the root, which is 2 for square roots). For example, $[sqrt{x^4} = x^2$, because $x^4$ is a perfect square.

With these principles in mind, we are well-equipped to tackle the given expression and identify its equivalent forms.

Deconstructing the Expression 5x√(28x⁵) + 8x³√(7x)

Our primary task is to simplify the expression $5x[sqrt{28x^5} + 8x^3[sqrt{7x}$. To do this effectively, we will break down each term individually, focusing on simplifying the radicals.

Step 1: Simplify the first term, $5x[sqrt{28x^5}$

• Begin by factoring the radicand, 28, into its prime factors: $28 = 2^2 \cdot 7$.
• Rewrite the radical as $5x[sqrt{2^2 \cdot 7 \cdot x^5}$.
• Now, separate the perfect square factors: $5x \cdot [sqrt{2^2} \cdot [sqrt{x^4} \cdot [sqrt{7x}$.
• Simplify the perfect squares: $5x \cdot 2 \cdot x^2 \cdot [sqrt{7x}$.
• Combine the coefficients and variables: $10x^3[sqrt{7x}$.

Step 2: Simplify the second term, $8x^3[sqrt{7x}$

• This term is already in a relatively simplified form, as the radicand, 7x, has no perfect square factors other than 1.
• Thus, the second term remains as $8x^3[sqrt{7x}$.

Step 3: Combine the simplified terms

• Now that we have simplified both terms, we can add them together:

  $$10x^3\[sqrt{7x} + 8x^3\[sqrt{7x}$$

• Since both terms have the same radical part, $x^3[sqrt{7x}$, we can combine their coefficients:

  $$(10 + 8)x^3\[sqrt{7x}$$

• This simplifies to $18x^3[sqrt{7x}$.

Therefore, the simplified form of the expression $5x[sqrt{28x^5} + 8x^3[sqrt{7x}$ is $18x^3[sqrt{7x}$. This simplified form will be our benchmark as we evaluate the expressions in the table.

Evaluating Expressions for Equivalence

Now that we have the simplified form, $18x^3[sqrt{7x}$, we can assess the given expressions in the table to determine which ones are equivalent. This involves simplifying each expression and comparing it to our benchmark.

Let's consider a hypothetical table with the following expressions:

We will now proceed to simplify each expression and determine its equivalence to $18x^3[sqrt{7x}$.

Expression 1: $5x[sqrt{28x} + 32x^3[sqrt{28x}$

• Simplify $[sqrt28x}$ $[sqrt{28x = [sqrt{2^2 \cdot 7x} = 2[sqrt{7x}$
• Substitute back into the expression: $5x(2[sqrt{7x}) + 32x^3(2[sqrt{7x})$
• Simplify: $10x[sqrt{7x} + 64x^3[sqrt{7x}$
• This expression cannot be directly combined into a single term that matches our benchmark, as it contains terms with different powers of x outside the radical.

Expression 2: $13x^4[sqrt{35x}$

• The radicand, 35x, has no perfect square factors other than 1. Therefore, the expression is already in its simplest form.
• Comparing $13x^4[sqrt{35x}$ to $18x^3[sqrt{7x}$, we see that they are not equivalent due to different coefficients, powers of x, and radicands.

Expression 3: $18x^3[sqrt{7x}$

• This expression is identical to our simplified form, making it equivalent.

Expression 4: $2x[sqrt{63x^5} + 4x^2[sqrt{28x3}$

• Simplify $[sqrt63x^5}$ $[sqrt{63x^5 = [sqrt{3^2 \cdot 7 \cdot x^4 \cdot x} = 3x^2[sqrt{7x}$
• Simplify $[sqrt28x^3}$ $[sqrt{28x^3 = [sqrt{2^2 \cdot 7 \cdot x^2 \cdot x} = 2x[sqrt{7x}$
• Substitute back into the expression: $2x(3x^2[sqrt{7x}) + 4x^2(2x[sqrt{7x})$
• Simplify: $6x^3[sqrt{7x} + 8x^3[sqrt{7x}$
• Combine like terms: $14x^3[sqrt{7x}$
• This expression is not equivalent to $18x^3[sqrt{7x}$ due to the different coefficient.

Expression 5: $9x^2[sqrt{28x2} - 2x[sqrt{7x^4}$

• Simplify $[sqrt28x^2}$ $[sqrt{28x^2 = [sqrt{2^2 \cdot 7 \cdot x^2} = 2x[sqrt{7}$
• Simplify $[sqrt7x^4}$ $[sqrt{7x^4 = x^2[sqrt{7}$
• Substitute back into the expression: $9x^2(2x[sqrt{7}) - 2x(x^2[sqrt{7})$
• Simplify: $18x^3[sqrt{7} - 2x^3[sqrt{7}$
• Combine like terms: $16x^3[sqrt{7}$
• This expression is not equivalent to $18x^3[sqrt{7x}$ because the radicand is different.

Summarizing Equivalent Expressions

After a thorough evaluation, we can confidently identify the expressions equivalent to $5x[sqrt{28x^5} + 8x^3[sqrt{7x}$. Based on our analysis, only Expression 3, which is $18x^3[sqrt{7x}$, is equivalent to the original expression.

This exercise underscores the significance of meticulous simplification and comparison when dealing with radical expressions. By understanding the fundamental principles of radical manipulation and applying them systematically, we can navigate complex expressions and unveil their underlying simplicity.