Problem 20 - Algebra A Combined Function

Question

Use radical notation to rewrite each expression. Simplify if possible. $$ (-8)^{4 / 3} $$

Step-by-Step Solution

Answer

The expression \((-8)^{\frac{4}{3}}\) simplifies to 16.

Step 1: Identify the Exponent Form

The expression is given as \[ (-8)^{\frac{4}{3}} \] This is in the form of \( a^{\frac{m}{n}} \), where \( a = -8 \), \( m = 4 \), and \( n = 3 \).

Step 2: Use Radical Notation

The expression \( a^{\frac{m}{n}} \) can be written in radical notation as \[ \sqrt[n]{a^m}. \] So, \[ (-8)^{\frac{4}{3}} = \sqrt[3]{(-8)^4}. \]

Step 3: Simplify the Radicand

Now, calculate \((-8)^4\). Since raising a negative number to an even power results in a positive number, \[ (-8)^4 = 4096. \] So the expression becomes \[ \sqrt[3]{4096}. \]

Step 4: Simplify the Cube Root

Find the cube root of 4096. This requires determining what number raised to the power of 3 gives 4096. After simplifying, we find that \[ \sqrt[3]{4096} = 16. \]

Step 5: Conclude the Simplification

The original expression \((-8)^{\frac{4}{3}}\) in radical notation is simplified to 16. Thus, \[ (-8)^{\frac{4}{3}} = 16. \]

Key Concepts

Exponents

Exponents are powerful mathematical tools that help express repeated multiplication of a number by itself. In the expression \((-8)^{\frac{4}{3}}\), the exponent is a fraction, specifically a rational exponent.
Here, the base is \(-8\), and the exponent \((\frac{4}{3})\) contains a numerator and a denominator. Understanding the exponent enables us to manipulate and simplify the expression using the appropriate mathematical operations.

• The numerator \(4\) tells us how many times to multiply the base \((-8)\) by itself.
• The denominator \(3\) represents the root we need to take after raising the base to the numerator.

Thus, \((-8)^{\frac{4}{3}}\) signifies finding the cube root of \((-8)^4\). Handling exponents requires recognizing how they describe both multiplication and division processes through the concept of roots. By translating the exponent into these steps, we reach further into the core of the expression.

Cube Root

The cube root is one of the key mathematical operations needed to understand and solve this expression. A cube root asks, "What number, when multiplied by itself three times, will equal the original number?"
In our exercise, we find the cube root of \(4096\), following simplifying \((-8)^4\). To achieve this, we compute \(\sqrt[3]{4096}\). This means looking for the number that results in \(4096\) when raised to the power of three.

• Calculating it manually involves seeing if smaller numbers \((such as 10, 12, 16)\) cubed approximate or exactly match \(4096\).
• Eventually, finding that \(16 \times 16 \times 16 = 4096\).

This shows that the cube root of \(4096\) is \(16\). Understanding cube roots helps decode expressions with fraction exponents by replacing multiplication with root finding, which simplifies the solution.

Simplification

Simplification is the process that takes complex expressions and makes them easier to work with by reducing them to their simplest form. In dealing with \((-8)^{\frac{4}{3}}\), we aim to make the expression more comprehensible and manageable.
The steps are:

• First, translate the fractional exponent into radical form, transforming \(a^{\frac{m}{n}} \/\rightarrow \/\sqrt[n]{a^m}\).
• Calculate any necessary powers, resulting in numbers like \(4096\).
• Apply root calculations to this value, finding simpler numbers that complete the equation.

Each step whittles down the complexity. Simplification eliminates exponential fractions through gradual transformation into integers, ensuring better clarity and ease for computational purposes.