Question

Challenge Problems. Perform the indicated operation and simplify. $$(2 x \sqrt{3 x})^{3}$$

Short answer

The simplified form is \(8 \cdot 3 \cdot \sqrt{3} \cdot x^{\frac{3}{2}}\), which can be further simplified to \(24 \cdot \sqrt{3} \cdot x^{\frac{3}{2}}\).

Step-by-step solution

Expand the Exponent

Apply the exponent of 3 to both the coefficient (2) and the term inside the square root. Remember that when raising a product to a power, you raise each factor in the product to that power separately.

Apply the Power to the Coefficient

Raise the coefficient 2 to the power of 3, which results in \(2^3 = 8\).

Apply the Power to the Square Root Term

Raise the contents of the square root to the power of 3. Since within the square root, you have \(3x\), you need to apply the power both to 3 and \(x\). This gives \((\sqrt{3x})^3 = (3x)^{\frac{3}{2}}\) because when you raise a square root to an exponent you multiply the exponents using the rule \((a^{m})^{n} = a^{m \cdot n}\).

Simplify the Raised Square Root Term

Since the exponent is \(\frac{3}{2}\), the '2' in the denominator signifies a square root, and '3' as the numerator is the power to which you raise the term. You can separate this as \(\sqrt{3x}^3 = (\sqrt{3})^3 \cdot (\sqrt{x})^3\). Simplify the square roots and the exponents: \((\sqrt{3})^3 = 3^{\frac{3}{2}} = \sqrt{3^3} = \sqrt{27}\) and \((\sqrt{x})^3 = x^{\frac{3}{2}}\).

Combine the Results

Multiply the simplified coefficient from Step 2 with the results from Step 4 to combine the terms into a single expression. You get \(8 \cdot \sqrt{27} \cdot x^{\frac{3}{2}}\).

Key concepts

Exponents and Radicals

Exponents and radicals are fundamental concepts in algebra that are closely related. An exponent represents repeated multiplication of a number by itself, while a radical, specifically a square root, signifies the number which, when multiplied by itself, yields the original number. In mathematical notation, exponents are written as a superscript, for example, in the expression \(a^n\), \(a\) is the base and \(n\) is the exponent. A radical is written with a root symbol, such as \(\sqrt{x}\), which represents the square root of \(x\).

When combining exponents and square roots, one must remember the property that \((a^m)^n = a^{m\cdot n}\). This is particularly helpful when simplifying expressions involving square roots raised to a power. Understanding the rule enables you to handle expressions where radicals are raised to an exponential power, as seen in the exercise \((2x\sqrt{3x})^3\), and manipulate them accurately to find a simplified form.

Operations with Square Roots

Operations with square roots, such as addition, subtraction, multiplication, and division, follow specific rules to maintain mathematical integrity. When dealing with multiplication, for instance, the product of two square roots can be combined into one square root containing the product of the radicands—the numbers under the roots—as expressed by the rule \(\sqrt{a} \cdot \sqrt{b} = \sqrt{ab}\).

When raising a square root to a power, as in the original exercise, it's key to understand how to apply the power to both the coefficient and the term inside the square root separately. This process can introduce exponents to the radicand, requiring further simplification, as with \((\sqrt{3x})^3\). The ability to perform these operations accurately is crucial in simplifying radical expressions to their most reduced form.

Simplifying Square Roots

Simplifying square roots involves reducing the expression under the root to its simplest form. One way to simplify square roots is by finding prime factors and pairing them according to the root's degree - for a square root, pairs of the same factor are sought. Factors outside the pair can be taken outside the radical sign, while perfect squares can be simplified directly—for example, \(\sqrt{4} = 2\).

In the case of the exercise \((2x\sqrt{3x})^3\), the exponent is applied across the expression, affecting both the numerical coefficient and the radical term. After this, the radical term is separated into individual square roots for simplification. This reveals perfect squares or enables the separation of terms for potential reduction, as with \(\sqrt{27}\), further leading to the simplification of the expression. Mastering these steps allows for clear and concise algebraic expressions, improving both understanding and computation accuracy.