Problem 10 - Algebra 2

Question

Simplify. Assume that all variables are positive. $$ \sqrt[3]{81 x^{2}} $$

Step-by-Step Solution

Answer

The resulting simplified expression is approximately \(4.33x^{\frac{2}{3}}\).

Step 1: Break down the cubic root

As per the root of a product rule, we can break down the cubic root expression into individual parts: \(\sqrt[3]{81}\) and \(\sqrt[3]{x^{2}}\).

Step 2: Find the cubic root of 81

To simplify \(\sqrt[3]{81}\), calculate the cubic root of 81. The cubic root of 81 is 4.326748710. For simplification purposes, we can approximate this to 4.33.

Step 3: Simplify the cube root of \(x^{2}\)

To simplify \(\sqrt[3]{x^{2}}\), remember that the cubic root of \(x^{2}\) is equivalent to \(x^{\frac{2}{3}}\).

Step 4: Combine results

Finally, combine the results of the cubic root of 81 and the cubic root of \(x^{2}\) to obtain the simplified form of the expression.

Key Concepts

Simplification

When we talk about simplification in mathematics, it means reducing an expression to its simplest form. This implies breaking down complicated parts into easier pieces you can manage. Simplification often involves combining like terms or reducing fractions.

• In our case, we simplify the expression by handling each component individually: \(81\) and \(x^2\).
• Each part of the expression \(\sqrt[3]{81 x^2}\) is treated separately under the cubic root.

Breaking down into manageable pieces helps us simplify and gain clarity on any mathematical expression.
It's a bit like tidying up a messy room. Once you have your parts simplified, it's easier to combine them back into a tidy whole.

Radical Expressions

Radical expressions involve roots, such as square roots or cube roots. A cube root, denoted as \(\sqrt[3]{x}\) for example, finds a value which, when multiplied by itself three times, equals \(x\). It's just like asking, "what number times itself three times is \(x\)?"

• In our problem, there's a cube root involved with both \(81\) and \(x^2\).
• Cube roots help in finding the root of the terms so we can simplify the expression further.

To handle cube roots, you can separate each component under the root and simplify individually. For numbers, it often involves finding an approximate value if it doesn’t simplify perfectly; for variables, you rewrite them using exponents with fractions. This way, radical expressions can be managed with ease.

Exponent Rules

Understanding exponent rules is vital for simplifying expressions with variables. Exponents indicate how many times a number is to be used as a factor.

• For cubing, \(x^{a/b}\) represents the \(b\)th root of \(x^a\).
• In the expression \(\sqrt[3]{x^2}\), it is rewritten as \(x^{\frac{2}{3}}\).

Exponent rules allow us to rewrite radicals in terms of exponents. They make it easier to manipulate and simplify expressions involving variables. Knowing these rules helps us reduce complex expressions and solve them effectively. So, whenever you see a square or cube root, remember you can often use exponent rules to help simplify!