Problem 37 - Technical Mathematics with Calculus

Question

Powers. Square the following expressions and simplify. $$4 \sqrt[3]{4 x^{2}}$$

Step-by-Step Solution

Answer

\( 16 \times \sqrt[3]{16 x^{4}} \)

Step 1: Squaring the expression

To square the given expression, which is a product of a constant and a cube root, we must square each factor separately. The expression can be written as \( (4)^(2) \times (\sqrt[3]{4 x^{2}})^{2} \).

Step 2: Squaring the constant

Square the constant 4. This gives us \( 4^{2} = 16 \).

Step 3: Squaring the cube root

Squaring the cube root of \(4x^{2}\) means we need to raise each factor within the root to the power of 2. We get \( (\sqrt[3]{4})^{2} \times (\sqrt[3]{x^{2}})^{2} \) which simplifies to \( \sqrt[3]{4^{2}} \times \sqrt[3]{x^{4}} \) or \( \sqrt[3]{16} \times \sqrt[3]{x^{4}} \).

Step 4: Combining the squared factors

Combine the constant squared with the squared cube root to get the final simplified expression. This can be written as \( 16 \times \sqrt[3]{16} \times \sqrt[3]{x^{4}} \). Combining cube roots we get \( 16 \times \sqrt[3]{16 x^{4}} \).

Key Concepts

Simplify Square Roots

Simplifying square roots involves reducing the expression under the square root symbol to its simplest form. To do this, one looks for factors that are perfect squares and separates them from those that aren't. When you take the square root of a perfect square, it results in a whole number. For instance, \(\sqrt{16}\) simplifies to 4 because 16 is a perfect square of 4. If you encounter \(\sqrt{32}\), you would break it down to \(\sqrt{16 \times 2}\), further simplifying it to \(4\sqrt{2}\) because \(\sqrt{16}\) equals 4.

In the exercise, squaring \(4\sqrt[3]{4x^{2}}\) didn't involve simplifying a square root directly, but similar principles apply when simplifying cube roots, which we'll explore in another section.

Exponent Rules

Exponent rules are the keys to simplifying expressions with powers. There are several important rules to keep in mind. The 'power of a power' rule states that for any non-zero number 'a' and integers 'm' and 'n', \(a^{m})^{n} = a^{mn}\). In the context of the exercise, squaring a cube root can be approached using this rule. For instance, \(\sqrt[3]{a})^{2}\) simplifies to \(\sqrt[3]{a^{2}}\) because you multiply the exponents.

Another vital rule is the 'product of powers' rule, which tells us that when multiplying like bases, you add the exponents: \(a^{m} \times a^{n} = a^{m+n}\). This rule is handy when combining squared factors with like bases, as seen in the final step of the exercise.

Cube Roots

Cube roots are a specific type of radical expression, representing the opposite operation of raising a number to the power of three. The cube root of a number 'a', denoted as \(\sqrt[3]{a}\), is the number 'b' such that \(b^{3} = a\). Simplifying cube roots involves identifying factors of the radicand (the number under the radical symbol) that are perfect cubes.

Simplify the Cube Root

Let's consider the expression \(\sqrt[3]{27}\), which simplifies to 3 because \(3^{3} = 27\). Similarly, if you see \(\sqrt[3]{8x^{3}}\), it simplifies to \(2x\) because both 8 and \(x^{3}\) are perfect cubes. In the original exercise, we had to square the cube root of \(4x^{2}\), but had it been a matter of simplification alone, we would look for a cube factor of 4, which does not exist, hence it remains under the cube root symbol.

Radical Expressions

Radical expressions contain roots, such as square roots or cube roots. The general form is \(\sqrt[n]{a}\), where 'n' is the index of the root and 'a' is the radicand. The process of simplifying involves finding factors of 'a' that are powers of 'n' and extracting them outside the radical.

Working with Radical Expressions

With variables, for example, simplifying \(\sqrt{x^{4}}\) results in \(x^{2}\) because the square of \(x^{2}\) is \(x^{4}\). When dealing with expressions containing both a constant and a variable, such as \(\sqrt{9x^{6}}\), we simplify to \(3x^{3}\) because 9 is a perfect square and \(x^{6}\) is a perfect square of \(x^{3}\).The original problem involved squaring radical expressions, which is closely related to the concept of simplifying since it involves manipulating the exponents within the roots. By squaring the cube root of \(4x^{2}\), we effectively applied exponent rules to radical expressions, illustrating their interconnectedness in mathematical practice.