Problem 31 - Algebra 2

Question

Simplify each number. $$64^{\frac{2}{3}}$$

Step-by-Step Solution

Answer

The simplified form of \(64^{\frac{2}{3}}\) is 16.

Step 1: Understanding the fractional exponent

The expression \(64^{\frac{2}{3}}\) indicates two operations because of the fractional exponent. The denominator '3' refers to taking a cube root, and the numerator '2' refers to squaring. The expression is equivalent to \((64^{1/3})^2\). This is because of the property of exponents that says \((a^m)^n = a^{mn}\).

Step 2: Compute the cube root

Now, calculate the cube root of 64, which is \(64^{1/3}\). The cube root of 64 is 4 because \(4*4*4 = 64\). Now, we have \((4)^2\).

Step 3: Squaring the result

Next, square the result from the previous step. Squaring 4 gives us 16.

Step 4: Simplifying to get the answer

Now, no further calculations are necessary and we see that the solution to \(64^{\frac{2}{3}}\) simplified is 16.

Key Concepts

Fractional Exponents

When dealing with fractional exponents, it's important to break them down into manageable parts. A fractional exponent consists of a numerator and a denominator. The denominator represents the root, while the numerator indicates the power or exponentiation to be performed after taking the root.

This is a key concept because it allows us to handle complex expressions more easily. For instance, in the example \(64^{\frac{2}{3}}\), the denominator '3' dictates that we need to find the cube root of 64 first. Then, the numerator '2' tells us to square the result of that cube root.

So, whenever you encounter a problem with fractional exponents, remember to:

• Identify the root by looking at the denominator.
• Apply the root to the base.
• Raise the resulting value to the power indicated by the numerator.

Approaching the problem this way ensures you handle each calculation step-by-step.

Properties of Exponents

Understanding the properties of exponents is crucial when simplifying expressions like \(64^{\frac{2}{3}}\). These properties offer rules that make complex calculations simpler and manageable. Let's discuss some useful properties:

• Product of Powers: \(a^m \cdot a^n = a^{m+n}\). When multiplying same bases, simply add exponents.
• Power of a Power: \((a^m)^n = a^{mn}\). This rule combines multiple layers of exponents into one calculation.
• Power of a Product: \((ab)^m = a^m \cdot b^m\). Apply the exponent to all parts inside the parenthesis.
• Zero Exponent: \(a^0 = 1\) for any non-zero 'a'. This helps simplify equations immediately.
• Negative Exponent: \(a^{-m} = \frac{1}{a^m}\). Inverts base into a fraction.

These fundamental properties are vital. In our original solution, we use the 'Power of a Power' property to convert \(64^{\frac{2/3}}\) into \((64^{1/3})^2\), showing the application of these properties in simplifying the expression step by step.

Cube Roots

Cube roots appear frequently especially in problems involving fractional exponents, and mastering them can greatly simplify such tasks. A cube root is the number that, when multiplied by itself twice, produces the original number.

In our solution, to simplify \(64^{\frac{2}{3}}\), we first find \(64^{1/3}\), which is the cube root of 64. This means finding a number that, when cubed (multiplied by itself two more times), gives 64.

• For \(64\), finding the cube root involves realizing that \(4 \times 4 \times 4 = 64\). Thus, \(64^{1/3} = 4\).

When faced with cube root problems:

• Use prime factorization if necessary to identify the cube root.
• Remember that cube roots can be determined by understanding the relationship between multiplication and exponentiation.

Certain numbers have cube roots that are readily recognizable, such as 8 (which has a cube root of 2), 27 (cube root 3), and of course, 64 (with a cube root of 4). Recognizing these makes solving fractional exponent problems far less daunting.