Problem 107 - Elementary and Intermediate Algebra

Question

The following problems involve addition, subtraction, and multiplication of radical expressions, as well as rationalizing the denominator. Perform the operations and simplify, if possible. All variables represent positive real numbers. $$ (10 \sqrt[3]{2 x})^{3} $$

Step-by-Step Solution

Answer

The simplified expression is \(2000x\).

Step 1: Understanding the Expression

The expression \((10 \sqrt[3]{2x})^3\) needs to be simplified. This involves raising the entire term inside the parentheses to the power of 3.

Step 2: Applying the Power of a Product Property

When you have a product inside an exponent, such as \((10 \sqrt[3]{2x})^3\), you can apply the power of a product rule, which means \((a \cdot b)^n = a^n \cdot b^n\). This simplifies the expression to \(10^3 \cdot (\sqrt[3]{2x})^3\).

Step 3: Evaluating \(10^3\)

Calculate \(10^3\) which means multiplying 10 by itself three times: \(10 \cdot 10 \cdot 10 = 1000\).

Step 4: Simplifying \((\sqrt[3]{2x})^3\)

The expression \((\sqrt[3]{2x})^3\) involves raising the cube root of \(2x\) to the third power. The cube root and the cube power are inverse operations, so \((\sqrt[3]{2x})^3\) simplifies to \(2x\).

Step 5: Multiply the Results

Now, multiply the results from Steps 3 and 4: \(1000 \cdot 2x = 2000x\).

Key Concepts

Addition of Radicals

Radical expressions are expressions that contain a root symbol, like square roots or cube roots. Adding radicals can sometimes be tricky because you need to ensure that the radicals you're adding are "like" radicals. "Like" radicals have the same index and the same radicand, which is the number or expression inside the radical sign. For example, \(\sqrt{2}\) and \(\sqrt{2}\) are like radicals, while \(\sqrt{2}\) and \(\sqrt{3}\) are not. You can add like radicals just like you add like terms, by adding their coefficients.

• For example: \(3\sqrt{2} + 5\sqrt{2} = 8\sqrt{2}\).

If the radicals are not "like," you cannot directly add them unless you can simplify them to become like radicals.

Subtraction of Radicals

Just like with addition, subtraction of radicals requires that the radicals be like terms. To subtract radicals, they must have the same index and radicand. If they do, you can subtract one radical's coefficient from the other just as you do with regular numbers.

• For example, \(7\sqrt{5} - 2\sqrt{5} = 5\sqrt{5}\).

If the radicals are not matching, try simplifying them to determine if they can become like radicals. If not, they must remain in their original form without subtracting.

Multiplication of Radicals

Multiplying radicals involves a few simple rules. You multiply the coefficients (the numbers outside the radical) and then multiply the radicands (the numbers inside the radical). This can often lead to opportunities for further simplification. For example, \(2\sqrt{3} \times 4\sqrt{6}\) would work as follows:

• Multiply the coefficients: \(2 \times 4 = 8\).
• Multiply the radicands: \(\sqrt{3} \times \sqrt{6} = \sqrt{18}\).
• The expression becomes \(8\sqrt{18}\), which can sometimes be simplified further depending on the radicand.

Always look for opportunities to simplify the resulting radical expression!

Rationalizing the Denominator

Having a radical in the denominator of a fraction is typically considered improper in math. Rationalizing the denominator involves removing the radical from the denominator, often by multiplying both the numerator and the denominator by a suitable expression. For example, consider the fraction \(\frac{5}{\sqrt{2}}\):

• Multiply both the numerator and the denominator by the conjugate of the denominator, which in this case is itself: \(\frac{5}{\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}}\).
• The denominator becomes \(\sqrt{2} \times \sqrt{2} = 2\) eliminating the radical.
• The expression simplifies to \(\frac{5\sqrt{2}}{2}\).

Rationalizing makes complex expressions easier to work with and is often required for a final answer in many math problems.