Problem 30 - Algebra 2

Question

Simplify each number. $$8^{\frac{2}{3}}$$

Step-by-Step Solution

Answer

The simplified form of \(8^{\frac{2}{3}}\) is \(4\).

Step 1: Raising to Power

The first step involves raising 8 to the power of the numerator of the fractional exponent. It's important to follow the parenthesis carefully because order of operations matters. Doing so gives us \(8^2 = 64\).

Step 2: Taking the Root

Next, the result from Step 1 is needed to be cube-rooted. This is done because the denominator of the fractional exponent implies taking the cube root. Therefore, \( \sqrt[3]{64} = 4\).

Step 3: Final Answer

After applying the operations indicated by the fractional exponent, we have arrived at the final answer, which is \(4\) in this case.

Key Concepts

Fractional Exponents

Fractional exponents can appear confusing at first, but they are essentially just another way to represent roots and powers on the same base number. When you see an expression like \(8^{\frac{2}{3}}\), it consists of two parts: the numerator \(2\) and the denominator \(3\). The numerator \(2\) tells you to square \(8\), while the denominator \(3\) instructs you to take the cube root after squaring the number. This dual action makes fractional exponents incredibly useful for simplifying complex expressions.

• Numerator: This is akin to the usual exponent that makes a base number larger. In our example, it squares the number \(8\), making it \(64\).
• Denominator: This signifies the root you take of the resulting number. Here, it is the cube root, reducing \(64\) back to a more manageable \(4\).

Understanding how the numerator and denominator operate can empower you to simplify expressions more confidently and watch out for potential calculation errors.

Order of Operations

The order of operations is a crucial concept in mathematics that dictates the sequence in which calculations should be performed to ensure accurate results. Remember the basic rule: PEMDAS, which stands for Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right).
When dealing with fractional exponents, it is important to first handle the exponentiation based on the numerator before taking the root indicated by the denominator. This ensures that you follow the proper sequence to avoid incorrect simplification. For example, in \(8^{\frac{2}{3}}\):

• Exponent: Raise \(8\) to the power of \(2\), resulting in \(64\).
• Root: Then take the cube root of \(64\) as indicated by the denominator \(3\), leading you to the correct final answer \(4\).

By adhering to these rules, complex calculations become a lot simpler to handle and less prone to mistakes.

Cube Roots

Cube roots are a type of root which simplifies calculations by bringing numbers back down to smaller values. The cube root of a number \(a\) is a number \(b\), which when multiplied by itself three times (\(b \times b \times b\)) gives back \(a\). For instance, the cube root of \(64\) is \(4\) because \(4 \times 4 \times 4 = 64\).
Cube roots often appear in fractional exponents, such as in \(8^{\frac{2}{3}}\). Following the steps properly, you take \(8\), raise it to the power of \(2\) to get \(64\), then take the cube root of \(64\) to simplify it to \(4\). This function of cube roots can help solve real-world problems, like determining volumes of cubes and more.

• Cubic calculations can also assist in understanding geometric concepts, especially in calculating volumes of cuboidal shapes.
• Always ensure to simplify the exponent first for an accurate result when involving cube roots.

Mastering cube roots can significantly enhance your ability to tackle complex mathematical challenges with ease.