Problem 92 - Algebra for College Students

Question

Explain how you would evaluate \(27^{\frac{2}{3}}\) without a calculator.

Step-by-Step Solution

Answer

The value of \(27^{\frac{2}{3}}\) is 9.

Step 1: Identify the Power and Root

The expression \(27^{\frac{2}{3}}\) can be split into a power and a root. This means you have to take the cube root of 27 first and then square the result. So, \(27^{\frac{2}{3}} = (27^{\frac{1}{3}})^2\).

Step 2: Find the Cube Root

Calculate \(27^{\frac{1}{3}}\), which is the cube root of 27. We know that \(3^3 = 27\), so \(\sqrt[3]{27} = 3\). Thus, \(27^{\frac{1}{3}} = 3\).

Step 3: Square the Result

Now that we have found the cube root, which is 3, we need to square it to complete our calculation. So calculate \(3^2 = 9\).

Step 4: Final Result

The evaluated expression \(27^{\frac{2}{3}}\) is equal to 9 after squaring the cube root of 27.

Key Concepts

Cube Root

The cube root is the inverse operation of raising a number to the power of three. In math, to find the cube root of a number means to figure out what number, when multiplied by itself three times, will equal the original number. This process is represented with the symbol \( \sqrt[3]{x} \). For example, the cube root of 27 can be determined because \( 3 \times 3 \times 3 = 27 \). Therefore, \( \sqrt[3]{27} = 3 \).

• The cube root "undoes" the cubing of a number.
• It can be found by determining the number that results in the original number when cubed.
• This is useful for simplifying expressions and solving equations involving powers of three.

Fractional Exponents

Fractional exponents can seem tricky, but they're simply another way of expressing roots alongside powers. When you see an expression like \( a^{m/n} \), it is equivalent to saying take the \( n^{th} \) root of \( a \) and then raise the result to the power of \( m \). In the example \( 27^{\frac{2}{3}} \), it means to find the cube root of 27 and then square it.
- The bottom number of the fraction (the denominator) indicates which root to take.- The top number (the numerator) tells you to which power to raise the result of the root.
This concept is highly valuable because it unites powers and roots under one notation, allowing us to easily understand complex expressions.

Power and Root Rules

The power and root rules in mathematics help simplify expressions that involve both powers and roots. These rules expand on the idea that powers and roots are inverse operations. Some basic guidelines are:

• \( (a^{m})^{n} = a^{m \times n} \). This rule simplifies expressions where two powers are used consecutively.
• \( \sqrt[n]{a} = a^{1/n} \). This relates rooted numbers with fractional exponents.
• Combining these, \( a^{m/n} = (a^{1/n})^{m} \), helps break down complex expressions into manageable parts.

Using these rules, we simplify \( 27^{\frac{2}{3}} \) by expressing it as \( (27^{1/3})^{2} \), highlighting the method to successively apply root and power operations. Mastery of these rules assists in tackling diverse algebraic problems involving exponents and roots.