Question

Simplify each expression. Assume that all variables are positive when they appear. $$\sqrt[3]{-40 x^{14} y^{10}}$$

Short answer

- \(\root{3}{40} \times x^{14/3} \times y^{10/3}\)

Step-by-step solution

Understand the Cube Root

The cube root of a number or expression, \(\root{3}\), is the value that, when multiplied by itself three times, gives the original number or expression.

Break Down the Expression

Rewrite the expression under the cube root: \(\root{3}{-40 x^{14} y^{10}}\)

Separate the Factors

Separate the expression into individual factors: \(\root{3}{-1 \times 40 \times x^{14} \times y^{10}}\).

Simplify the Constant

Simplify the constant: \(\root{3}{-1} \times \root{3}{40}\). Since \(\root{3}{-1} = -1\), the expression becomes: - \(\root{3}{40}\)

Simplify the Variables

Simplify the variables separately: \[ \root{3}{x^{14}} \times \root{3}{y^{10}} \]. Each can be simplified using the property \(a^{m/n} = \root{n}{a^m}\).

Apply Exponent Rules

Rewrite using exponent rules for cube roots: \((x^{14})^{1/3} = x^{14/3}\) and \((y^{10})^{1/3} = y^{10/3}\).

Combine Simplified Parts

Combine all the simplified parts: \(- \root{3}{40} \times x^{14/3} \times y^{10/3} \).

Key concepts

Cube Root

The cube root of a number or expression, denoted as \(\root{3}\), is a special operation that finds the value which, when multiplied by itself three times, gives the original number. For example, the cube root of 27 is 3 because \(3 \times 3 \times 3 = 27\). The cube root can also be applied to algebraic expressions, helping to simplify complex terms.

Exponent Rules

Exponent rules are essential when working with algebraic expressions. They help simplify expressions and perform calculations more efficiently. Key rules include:

Factorization

Factorization is the process of breaking down a complex expression into simpler factors that, when multiplied together, yield the original expression. In our example, breaking down the expression \(\root{3}{-40 x^{14} y^{10}}\) involves identifying individual factors like \(-1, 40, x^{14},\) and \(y^{10}\). This makes it easier to apply other simplification methods such as exponent rules and cube root calculations.

Simplifying Variables

Simplifying variables in expressions involves using exponent rules to make the expression more manageable. For example, if we have \(x^{14}\), we can simplify \((x^{14})^{1/3}\) using the exponent rule \((a^m)^n = a^{mn}\), resulting in \(x^{14/3}\). Similarly, for \(y^{10}\), \((y^{10})^{1/3}\) becomes \(y^{10/3}\). Combining these simplified variables with constants yields a more concise and simpler expression: \(- \root{3}{40} \times x^{14/3} \times y^{10/3}\).