Problem 50 - Algebra 2

Question

Simplify each number. $$(-343)^{\frac{1}{3}}$$

Step-by-Step Solution

Answer

-7

Step 1: Understand the fraction as an exponent

First, note that the exponent \(\frac{1}{3}\) signifies a cube root. This means that we are looking for a number that, when cubed, equals -343.

Step 2: Identify the cube root

Knowing that the cube root of 343 is 7, we should recognize that the cube root of -343 is -7. This is because the cube of 7, \(7^3\), is 343, and the cube of -7, \((-7)^3\), is -343.

Step 3: Substitute back into the expression

Now that we know the cube root of -343 is -7, we can simplify \( (-343)^{\frac{1}{3}} \) to be -7.

Key Concepts

Cube Roots

Cube roots are the inverse operation of cubing a number. When we calculate the cube root of a number, we are essentially looking for a value that, when multiplied by itself three times, gives the original number. For example, the cube root of 8 is 2, because \[2 \times 2 \times 2 = 8.\]This concept also applies to negative numbers. The cube root of \((-343)\) is \(-7\), because:

• \((-7) \times (-7) = 49\)
• \(49 \times (-7) = -343\)

This highlights that cube rooting a negative number will result in a negative root. It is important to note that not every type of root permits negative values under the radical, but the cube root is an exception. Its ability to work with negative inputs makes it quite versatile.

Simplifying Expressions

Simplifying expressions with rational exponents requires an understanding of how exponents and roots interact. A rational exponent is a fraction like \(\frac{1}{3}\), which indicates a root. The process of simplifying involves converting the expression to a root, calculating the root, and rewriting the expression with a simplified result.For the expression \((-343)^{\frac{1}{3}}\):

• The exponent \(\frac{1}{3}\) tells us to find the cube root of \(-343\).
• We know from experience that the cube root of \(-343\) is \(-7\).
• Therefore, the expression simplifies directly to \(-7\).

Breaking down the expression into these manageable parts not only simplifies the calculation but also provides a clear pathway from the initial expression to the final simplified form.

Negative Numbers

Handling negative numbers in mathematical operations can seem tricky at first, but understanding their properties makes it easier. When you cube a negative number, the result is always negative because:

• A negative times a negative is positive (e.g., \(-3 \times -3 = 9\)).
• Multiplying this positive result by another negative makes it negative again (e.g., \(9 \times -3 = -27\)).

In the exercise, we handle a negative number under a radical with a rational exponent, specifically \((-343)^{\frac{1}{3}}\). Because the operation of cube rooting allows us to keep the negative sign, the resulting root \(-7\) is negative.These inherent properties of negative numbers are crucial for correctly performing operations that involve them, ensuring that our expressions are simplified accurately.