Problem 14 - Algebra for College Students

Question

Evaluate each numerical expression. \(64^{\frac{2}{3}}\)

Step-by-Step Solution

Answer

The evaluated expression is 16.

Step 1: Understand the expression

The expression provided is a power expression with a fractional exponent: \( 64^{\frac{2}{3}} \). The base is \( 64 \) and the exponent is \( \frac{2}{3} \). This expression signifies both a root and a power.

Step 2: Break down the exponent

The exponent \( \frac{2}{3} \) can be interpreted as finding the cube root and then squaring the result. This means \( 64^{\frac{2}{3}} = (64^{1/3})^2 \).

Step 3: Find the cube root of the base

Calculate \( 64^{1/3} \). The cube root of \( 64 \) is \( 4 \) because \( 4^3 = 64 \). So, \( 64^{1/3} = 4 \).

Step 4: Square the result of the cube root

Now square the result of the cube root: \((64^{1/3})^2 = 4^2 \). Calculate \( 4^2 \), which equals \( 16 \).

Step 5: Conclude the evaluation

The evaluated result of \( 64^{\frac{2}{3}} \) is \( 16 \). Therefore, \( 64^{\frac{2}{3}} = 16 \).

Key Concepts

cube root

A cube root of a number refers to a special value that, when multiplied by itself three times, gives the original number. For example, the cube root of 64 is 4, shown by the equation: - \(4 \times 4 \times 4 = 64\). When you see a fractional exponent with a denominator of 3, it often indicates that a cube root operation is involved. Understanding this allows you to break more complex expressions into simpler parts. It's like the opposite of cubing a number.

power expression

A power expression consists of a base and an exponent. In a power expression like \(64^{\frac{2}{3}}\), 64 is the base and \(\frac{2}{3}\) is the exponent. These expressions show how many times a number, the base, is multiplied by itself. The challenge in fractional exponents lies in interpreting the combination of power and root operations represented by the fraction.

fractional exponent

Fractional exponents, such as \(\frac{2}{3}\), include both a power and a root operation. The numerator (2) is the power, while the denominator (3) indicates the root. A fractional exponent transforms an ordinary power expression by adding the complexity of a radical operation. You tackle it by first breaking down the exponent into a root operation (for the denominator) followed by a power operation (for the numerator).

• First, find the root of the base \(b^{1/n}\)
• Then, apply the power \((b^{1/n})^m\).

exponentiation

Exponentiation is the mathematical operation of raising a base to the power of an exponent. This operation is vital for simplifying complex expressions, like those with fractional exponents. Exponentiation increases the base by multiplying it by itself a number of times equal to the exponent. In \(64^{\frac{2}{3}}\), exponentiation and root operations combine, requiring precise steps starting from finding the cube root and ending with squaring the resulting value.