Problem 27 - Intermediate Algebra

Question

Find each cube root. $$ \sqrt[3]{-27 x^{9}} $$

Step-by-Step Solution

Answer

The cube root is \(-3x^3\).

Step 1: Understand the Cube Root

The cube root of a number or expression, \( \sqrt[3]{a} \), is the value that must be multiplied by itself three times to give \( a \). Here, we need to find the cube root of the expression \( -27x^9 \).

Step 2: Separate Numerical and Variable Parts

Separate the numerical part from the variable part in \( \sqrt[3]{-27x^9} \). Thus, consider \( \sqrt[3]{-27} \) and \( \sqrt[3]{x^9} \).

Step 3: Calculate the Cube Root of the Numerical Part

The cube root of \( -27 \) is \( -3 \). This is because \( (-3) \times (-3) \times (-3) = -27 \).

Step 4: Calculate the Cube Root of the Variable Part

For \( \sqrt[3]{x^9} \), use the property of exponents: \( \sqrt[3]{x^9} = x^{9/3} = x^3 \).

Step 5: Combine the Results

Combine the results from steps 3 and 4: \( \sqrt[3]{-27x^9} = (-3) \times x^3 = -3x^3 \).

Key Concepts

Cube Root Calculation

The process of cube root calculation involves determining a number which, when multiplied by itself three times, results in the original value. For the expression \( \sqrt[3]{a} \), you are looking for a number \( b \) such that \( b^3 = a \). Calculating cube roots might involve both numerical values and variable expressions.

For example, if we have the expression \( \sqrt[3]{-27x^9} \), we first consider the cube root of the numerical part \(-27\) and the variable part \(x^9\) separately. The aim is to identify a simpler form of the original expression.

• If the expression has negative numbers, remember that a cube root of a negative number remains negative because multiplying an odd number of negative factors gives a negative result.
• Recognize that cube roots can simplify expressions, reducing complex calculations into more manageable results.

Exponents

Exponents represent repeated multiplication of a number by itself. For example, \(x^9\) signifies that \(x\) is multiplied by itself nine times. Exponents are crucial in simplifying expressions, particularly when dealing with roots.

Understanding the property of exponents with respect to roots is essential. Using the rule \(a^{m/n} = \sqrt[n]{a^m}\), you can simplify many mathematical expressions. In the context of cube roots, the exponent rule states, \( \sqrt[3]{x^9} = x^{9/3} = x^3 \).

• Each aspect of multiplying exponents operates under clearly defined arithmetic rules.
• Knowing how to handle exponents helps streamline calculations involving expressed roots.

Variable Expressions

Variable expressions involve algebraic terms that include variables such as \(x\), \(y\), or other symbols used to represent numbers. In the example \( \sqrt[3]{x^9} \), \(x\) is a variable, and you're asked to determine its cube root.

Breaking down a variable expression into simpler terms is the key to solving it effectively. In our instance, we calculate the cube root of \(x^9\) by rewriting it as \(x^{9/3} = x^3\). This signifies that \(x\) is being raised to the power of 3, reinforcing our understanding of exponents and roots.

• This process makes the expressions much easier to handle, as they become simple multiplications or divisions rather than large or tangled equations.
• Such conversions are beneficial while dealing with more complex algebraic questions.

Root Properties

Root properties are mathematical rules that assist in breaking down or combining roots effectively. The most applicable property here is root multiplicity. The cube root of a multiplied expression \( \sqrt[3]{ab} \) is the same as \( \sqrt[3]{a} \times \sqrt[3]{b} \). Applying this property simplifies the calculation by dealing with each factor separately.

This approach worked well when we considered \( \sqrt[3]{-27x^9} \) because \( \sqrt[3]{-27} \) and \( \sqrt[3]{x^9} \) were computed independently, allowing their multiplication to give the final result \(-3x^3\).

• Utilizing root properties helps in reducing intricate root expressions to more straightforward forms.
• Understanding these properties ensures clarity and accuracy in mathematical solutions, reducing the risk of mistakes with large computations.