Problem 68 - Algebra 2

Question

Multiply and simplify. Assume that all variables are positive. $$ -5 \sqrt{6 x} \cdot 3 \sqrt{6 x^{2}} $$

Step-by-Step Solution

Answer

The simplified expression for \(-5 \sqrt{6x} \cdot 3 \sqrt{6x^{2}}\) is \(-90x\sqrt{x}\).

Step 1: Multiply coefficients

Multiplication starts with the coefficients, -5 and 3. So, -5 times 3 equals -15.

Step 2: Combine and simplify square root terms

Now, look at the multiplication of \(\sqrt{6x}\) and \(\sqrt{6x^{2}}\). According to the property of square roots, \(\sqrt{a} * \sqrt{b} = \sqrt{ab}\), it can be simplified as \(\sqrt{(6x)*(6x^{2})}\). This equals \(\sqrt{36x^{3}}\). Since 36 is a perfect square, it can be simplified to 6 and \(x^{3}\) simplifies to \(x\sqrt{x}\). So, \(\sqrt{36x^{3}}\) equals to \(6x\sqrt{x}\)

Step 3: Combine results

Now, combine the result from Step 1 with that from Step 2 to get the final result. That is, -15 times \(6x\sqrt{x}\), which is \(-90x\sqrt{x}\).

Key Concepts

Simplifying Radicals

Simplifying radicals is a key component of algebra that involves expressing a radical expression in its simplest form. Radicals often appear in the form of square roots, cube roots, and so on. In this exercise, we mainly focus on square roots.

To simplify a radical, you need to identify factors of the number under the square root that are perfect squares. A perfect square can be removed from under the square root, simplifying the expression. For example, consider the expression \( \sqrt{36} \). Since 36 is a perfect square (\(6^2\)), it simplifies to 6.

When dealing with variables, simplifying radicals also involves reducing the power of the variable under the radical. Using our exercise's example: \( \sqrt{x^3} \) simplifies to \( x \sqrt{x} \) because \( x^3 = x^2 \times x \) and we can take \( x^2 \) out of the radical as \( x \).

**Tips for Simplifying Radicals:**

• Identify and extract perfect squares or powers from the radical.
• Apply similar simplification rules to the coefficients and the variables.
• Rewrite the expression by multiplying any constants and variables taken out of the radical.

Properties of Square Roots

Understanding the properties of square roots is crucial in simplifying radical expressions and performing operations like multiplication of radicals. These properties provide us with rules that help simplify our work significantly.

One of the important properties used in this exercise is \( \sqrt{a} \times \sqrt{b} = \sqrt{ab} \). This property allows us to combine two or more radicals into one single radical. In the problem's solution, the square roots \( \sqrt{6x} \) and \( \sqrt{6x^2} \) are multiplied to become \( \sqrt{36x^3} \).

Another useful property for simplification is \( \sqrt{x^n} = x^{n/2} \), which allows you to transform a radical expression involving powers into an equivalent rational exponent form, making calculations easier.

**Key Properties to Remember:**

• \( \sqrt{a} \times \sqrt{b} = \sqrt{ab} \)
• \( \sqrt{a^2} = a \) if \( a \geq 0 \)
• \( \sqrt{x^n} = x^{\frac{n}{2}} \)

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operations that stand for a particular value or set of values. These expressions can be manipulated through various algebraic rules and operations, such as addition, subtraction, multiplication, division, and more complex functions like exponentiation and root extraction.

In the context of multiplying radicals, algebraic expressions may include terms that are both coefficients and radicals. For example, in our problem, the expression \( -5 \sqrt{6x} \cdot 3 \sqrt{6x^2} \) involves both numerical coefficients (\(-5\) and \(3\)) and radical expressions (\(\sqrt{6x}\) and \(\sqrt{6x^2}\)).

Simplifying such expressions often requires combining like terms and using properties of numbers and radicals to express the result in its simplest form. When multiplying the coefficients, as in step 1 of the solution, we end up with \(-15\). Further combining with the simplified radical from step 2, we get the final expression \(-90x\sqrt{x}\).

**Key Ideas for Algebraic Expressions:**

• Combine coefficients separately from the radical expressions.
• Apply properties of operations consistently to obtain the simplest form.
• Ensure proper expression of algebraic results, respecting variable positivity assumptions.