Problem 11 - Algebra 2

Question

Simplify. $$ 3 \sqrt[3]{81}-2 \sqrt[3]{54} $$

Step-by-Step Solution

Answer

The simplified expression is \(9\cdot\sqrt[3]{3}-6\cdot\sqrt[3]{2}\)

Step 1: Rewrite as powers of 3

The numbers inside the cubic roots can be rewritten as powers of 3. Specifically, \(81 = 3^4\) and \(54 = 2\cdot 3^3\). So we write the expression as 3\(\sqrt[3]{3^4}\) - 2\(\sqrt[3]{2\cdot 3^3}\)

Step 2: Simplify cubic roots

Using the property that \(\sqrt[3]{a^3} = a\), we can simplify each of the cubic roots. We get \(3\cdot 3^{4/3} - 2\cdot 2^{1/3} \cdot 3\)

Step 3: Simplify further

Next, simplify \(3^{4/3}\) by writing it as \(3^{1+1/3}\), which equals \(3\cdot 3^{1/3}\). This changes the expression to \(3^2\cdot 3^{1/3} - 6^{1/3}\cdot 3\).

Key Concepts

Cubic Roots

Cubic roots are mathematical operations that find a number which, when multiplied by itself three times, gives the original number. This is the third root of a number, and it's represented as \( \sqrt[3]{x} \). Cubic roots are useful in simplifying expressions, especially when dealing with terms that need to be expressed in simpler forms.

• For example, the cubic root of 27 is 3, because \( 3^3 = 27 \).
• The cubic root operation can also be expressed as a power: \( x^{1/3} = \sqrt[3]{x} \).

When simplifying expressions like \( 3 \sqrt[3]{81} - 2 \sqrt[3]{54} \), first identify numbers inside the cubic roots and attempt to express them as cubes of simpler numbers.
This makes it easier to apply the cubic root properties and simplifies the entire expression.

Exponent Laws

Exponent laws are rules that guide how to manipulate powers and simplify expressions involving exponents. These laws come into play when you're rewriting or simplifying expressions.
Some key exponent laws include:

• Product of Powers: \( a^m \cdot a^n = a^{m+n} \)
• Power of a Power: \( (a^m)^n = a^{m \cdot n} \)
• Quotient of Powers: \( a^m / a^n = a^{m-n} \)
• Power of a Product: \( (ab)^m = a^m \cdot b^m \)

Applying exponent laws can be particularly helpful when you're looking to simplify complex expressions. For instance, if you have \( 3^{4/3} \) in an expression, recognize that it can be split into \( 3^1 \cdot 3^{1/3} \) using the power of a product rule.
Understanding these basic rules allows for a thorough simplification of expressions involving powers, which is often necessary in exercises involving cubic roots.

Radical Expressions

Radical expressions involve roots, such as square roots, cube roots, and higher-order roots. The root symbol \( \sqrt{} \) indicates a radical expression, and the number inside the root sign is called the radicand.
Radical expressions can sometimes be simplified by factoring the radicand into a form that makes it easy to extract roots. For example, \( \sqrt[3]{81} \) can be simplified because 81 can be written as \( 3^4 \).
To further simplify radical expressions, you can apply the following techniques:

• Ensure the radicand is expressed in its simplest form, often requiring factoring into prime numbers.
• Use the property \( \sqrt[n]{a^m} = a^{m/n} \) to simplify expressions.
• Combine like terms when possible to further reduce the expression's complexity.

Simplifying radical expressions often involves identifying common factors or rewriting parts of the expression using exponentiation, allowing for easier manipulation and understanding of quantitative relationships.