Problem 63 - Algebra 2

Question

Simplify each expression. Rationalize all denominators. Assume that all variables are positive. $$ \sqrt[3]{3} \cdot \sqrt[3]{18} $$

Step-by-Step Solution

Answer

The simplified form of \( \sqrt[3]{8000} \) is 20.

Step 1: Express as cubes

Start by expressing 8000 as a product of cubes. 8000 can be expressed as \(20^3\). So, the cube root should be \( \sqrt[3]{20^3} \)

Step 2: Apply cube root

Now, apply the cube root to the expression. The cube root of a number's cube is the number itself, so \( \sqrt[3]{20^3} \) simplifies to 20.

Key Concepts

Cube Roots

Understanding cube roots is an essential part of simplifying expressions. A cube root is an inverse operation to cubing a number. When you see a cube root, it means finding a number which, when multiplied by itself three times, gives the original number inside the root. This operation is symbolized by \( \sqrt[3]{x} \). For instance, the cube root of 8000 involves finding a number that, when cubed, equals 8000.
If you express 8000 as \( 20^3 \), what you're really doing is identifying that \( 20 \times 20 \times 20 = 8000 \). Therefore, \( \sqrt[3]{8000} \) simplifies directly to 20 because the cube root and cube are inverse operations that undo each other.
This simplification helps in making expressions more manageable and numbers easier to work with.

Rationalizing Denominators

In mathematics, especially when dealing with roots, you might encounter expressions with roots in the denominator. Rationalizing a denominator is the process of eliminating any root or radical from the bottom part of a fraction. This way, the expression only has integer values or simple fractions in the denominator.
To rationalize a denominator with cube roots or any other type of root, you can multiply the numerator and the denominator by a quantity that will eliminate the root when multiplied out. However, for expressions involving only numerical cube roots, like our simplified problem, this step wasn't necessary because we don't have a fraction or root in the denominator to deal with.
Understanding this concept is vital because it ensures that all parts of your expression are in their simplest and most standard form.

Exponents

Exponents are a way of representing repeated multiplication of a number by itself. Specifically, an exponent tells you how many times to multiply the base by itself. For example, \( 20^3 \) implies multiplying 20 by itself three times: \( 20 \times 20 \times 20 \).
Exponents follow specific rules, such as power of a power, product of powers, and quotient of powers, each serving to simplify expressions neatly. In our exercise, knowing that \( 8000 = 20^3 \), allows us to handle its cube root directly and efficiently.

• Power of a power rule: \((a^m)^n = a^{m\times n}\)
• Product of powers rule: \(a^m \times a^n = a^{m+n}\)
• Quotient of powers rule: \(\frac{a^m}{a^n} = a^{m-n}\)

These rules are key to simplifying expressions and solving problems with exponents effectively.

Positive Variables

When simplifying expressions, especially those involving roots and exponents, the assumption is often made that variables are positive. This simplifies the process, as positive numbers behave predictably with roots and powers.
For example, if \( x \) is positive, \( \sqrt[3]{x^3} = x \). This is straightforward because cube roots and cubes negate each other without concerns about negative values, which can affect the outcome. Working with negative numbers involves additional rules and precautions.
Assuming variables are positive ensures that the simplified forms of the expressions are valid across all applicable mathematical principles, perfectly aligning with the problem's instructions to handle only positive variables for simplicity and consistency.