Problem 5 - Algebra 2

Question

Simplify each expression. $$(-3)^{\frac{1}{3}} \cdot(-3)^{\frac{1}{3}} \cdot(-3)^{\frac{1}{3}}$$

Step-by-Step Solution

Answer

The simplified expression is -27

Step 1: Calculation of cube root

The cube root of -3, \((-3)^{\frac{1}{3}}\) equals to -3. So the given expression \( (-3)^{\frac{1}{3}} \cdot(-3)^{\frac{1}{3}} \cdot(-3)^{\frac{1}{3}}\) transforms into -3 \cdot -3 \cdot -3.

Step 2: Multiplication

The product of -3 \cdot -3 \cdot -3 equals to -3 cubed, which equals to -27.

Step 3: Final Result

After performing these steps, the expression is simplified to -27

Key Concepts

Cube Root

When we talk about cube roots, we are looking for a number that, when multiplied by itself three times, gives us the original number.
In mathematical terms, the cube root of a number \(x\) is represented as \(x^{\frac{1}{3}}\).
For instance, if we have \((-3)^{\frac{1}{3}}\), we are in search of a value that can be multiplied thrice to get back to \(-3\).
It’s important to note that cube roots of negative numbers are real numbers. This is because multiplying an odd number of negative numbers results in a negative sign.
Thus, the cube root of \(-3\) is simply \(-3\), since \(-3) \times (-3) \times (-3) = -3^3 = -27\).

Simplifying Expressions

The process of simplifying expressions involves making them easier to work with by combining like terms or reducing to the most basic form.
In this particular exercise, you start with the expression \((-3)^{\frac{1}{3}} \cdot (-3)^{\frac{1}{3}} \cdot (-3)^{\frac{1}{3}}\).
Instead of calculating each cube root separately and then multiplying, recognizing the properties of exponents can simplify it.

• First, note that products of powers with the same base are added together: \((-3)^{\frac{1}{3}} \times (-3)^{\frac{1}{3}} \times (-3)^{\frac{1}{3}} = (-3)^{1}\).
• This means that the original expression simplifies directly to \(-3\).

By understanding these steps, you reduce a complex expression into something more manageable.

Multiplication of Numbers

Multiplication is one of the basic arithmetic operations, and it's all about adding a number to itself a specified number of times.
When dealing with multiplication, especially with negative numbers, keep these points in mind:

• Multiplying a negative number by another negative number gives a positive result.
• However, multiplying a positive and a negative number results in a negative product.
• The product of three negative numbers remains negative, as in the case of \(-3 \times -3 \times -3\), which equals \(-27\).

Understanding these rules helps simplify expressions and avoid confusion when calculating products of both positive and negative numbers.