The cube root is a mathematical operation that finds the value that, when multiplied by itself three times, results in the original number. It is the inverse operation of raising a number to the power of three.
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The cube root of a number is denoted by the symbol $\sqrt[3]{}$, where the 3 indicates that the root being taken is the cube root.
Cube roots are used to solve problems involving the volume of three-dimensional shapes, such as cubes and rectangular prisms.
Cube roots can be used to simplify expressions with variables raised to the power of 3, by rewriting them using the cube root symbol.
When adding, subtracting, or multiplying radical expressions with cube roots, the cube roots can be combined using the properties of exponents.
Solving radical equations that involve cube roots requires isolating the cube root term and then using the inverse operation (raising both sides to the power of 3) to solve for the variable.
Review Questions
How can the cube root be used to simplify expressions with variables raised to the power of 3?
To simplify an expression with a variable raised to the power of 3, the cube root can be used. For example, the expression $x^3$ can be rewritten as $\sqrt[3]{x^3}$, which is simply $x$. This allows for easier manipulation and simplification of the expression.
Describe the process of adding, subtracting, or multiplying radical expressions with cube roots.
When adding, subtracting, or multiplying radical expressions with cube roots, the cube roots can be combined using the properties of exponents. For example, to add $\sqrt[3]{a}$ and $\sqrt[3]{b}$, the result would be $\sqrt[3]{a + b}$. Similarly, to multiply $\sqrt[3]{a}$ and $\sqrt[3]{b}$, the result would be $\sqrt[3]{a \cdot b}$. This allows for the simplification of complex radical expressions involving cube roots.
Explain how cube roots can be used to solve radical equations.
Solving radical equations that involve cube roots requires isolating the cube root term and then using the inverse operation (raising both sides to the power of 3) to solve for the variable. For example, to solve the equation $\sqrt[3]{x} = 5$, we would raise both sides to the power of 3, resulting in $x = 125$. This process can be extended to more complex radical equations, allowing for the determination of the variable's value.