Problem 73 - College Algebra with Modeling and Visualization

Question

Simplify the expression. Assume that all variables are positive. $$ 2 \sqrt[3]{16}+\sqrt[3]{2}-\sqrt{2} $$

Step-by-Step Solution

Answer

Simplified expression: \(2^{7/3} + 2^{1/3} - 2^{1/2}\)

Step 1: Convert to Exponential Form

Convert the cube roots and square root into exponential form: \[ \sqrt[3]{16} = 16^{1/3}, \quad \sqrt[3]{2} = 2^{1/3}, \quad \text{and} \quad \sqrt{2} = 2^{1/2} \]

Step 2: Simplify the Cube Root of 16

Express 16 as a power of 2:\[ 16 = 2^4 \]Thus,\[ \sqrt[3]{16} = (2^4)^{1/3} = 2^{4/3} \]

Step 3: Substitute Simplified Terms Back

Substitute the exponential forms into the original expression:\[ 2 \times 2^{4/3} + 2^{1/3} - 2^{1/2} \]

Step 4: Simplify the Expression

Combine like terms:\[ 2^{1} \times 2^{4/3} = 2^{3/3 + 4/3} = 2^{7/3} \]So the expression becomes:\[ 2^{7/3} + 2^{1/3} - 2^{1/2} \]

Key Concepts

Exponential Form

Converting roots into exponential form is an essential step in algebra. It helps simplify expressions more easily. In exponential notation, a root is represented as a fractional exponent.

For example, a square root \( \sqrt{x} \) can be written as \( x^{1/2} \). Similarly, a cube root \( \sqrt[3]{x} \) is expressed as \( x^{1/3} \). This method allows us to utilize the laws of exponents to simplify expressions.

Using exponential form, we can easily multiply and divide roots. It's done through the basic property of exponents: \( a^m \times a^n = a^{m+n} \). Applying these rules to the conversion of roots into exponential form makes algebraic manipulation straightforward. This approach is especially useful when dealing with roots of the same base number.

Cube Roots

Cube roots represent a value that, when multiplied by itself three times, gives the original number. Simplifying a cube root involves finding a number raised to one third power.

To simplify cube roots, recognize if the number can be expressed as a power of another number. For example, 16 can be expressed as \( 2^4 \). Therefore, its cube root is \( (2^4)^{1/3} = 2^{4/3} \).

When working with cube roots:

• Try expressing the number inside the root as a power of a suitable base number.
• Convert the cube root into exponential form for easier manipulation.
• Use the property \( (a^m)^n = a^{m \times n} \) to solve.

Cube roots can be simplified further by combining terms if they share the same base and exponent form.

Square Roots

The square root of a number is a value that, when squared, returns the original number. It can be rewritten in exponential form, which is useful for simplification.

For instance, \( \sqrt{2} \) becomes \( 2^{1/2} \). This fractional exponent format helps in algebraic operations, as you can apply it alongside other exponential terms.

When simplifying square roots:

• Convert the root to exponential form to simplify the manipulation.
• Combine with other roots using properties of exponents like addition or subtraction.
• Check for perfect squares first, as they can simplify directly.

Utilizing square roots in exponential form allows for seamless simplification in more complex expressions and helps work with algebraic formulas more effectively.