Problem 43 - BIG IDEAS MATH Algebra 2: Common Core Student Edition 2015

Question

Evaluate the expression without using a calculator. $$ 27^{2 / 3} $$

Step-by-Step Solution

Answer

The evaluated result of the expression \(27^{2/3}\) is 9.

Step 1: Interpreting the fractional exponent

As per the rule of fractional exponents, \(a^{n/m} = root \: m \: of (a^n)\), the given expression \(27^{2/3}\) becomes cube root of \(27^2\).

Step 2: Evaluate the exponent

Evaluating 27^2 yields 729. So our expression becomes cube root of 729.

Step 3: Evaluate the Cube root

Finally, the cube root of 729 is 9 as \(9^3 = 729\).

Key Concepts

Evaluating Expressions

When we are evaluating expressions with exponents, particularly those with fractional exponents, understanding the order of operations is crucial. Evaluating expressions is more than just calculating; it entails interpreting the components of the expression and accurately applying mathematical rules to simplify or find its value.

To evaluate an expression like \(27^{2/3}\), one must recognize that the fraction indicates not only a power but also a root, a process that is a key aspect of understanding exponentiation in algebra. Evaluating expressions with fractional exponents is simplified by first resolving the exponent part, then applying the root.

Cube Root

Understanding the cube root of a number is like asking, 'which number, when multiplied by itself three times, gives the original number?' This root is signified by the radical symbol with a little three above it, or in fractional exponent terms, as an exponent of \(1/3\).

For instance, the cube root of 8 is 2, because \(2 \times 2 \times 2 = 2^3 = 8\). In our exercise, determining the cube root is part of the task. The cube root of 729 is crucial to find because it is the last step to evaluate the original expression \(27^{2/3}\).

Exponentiation Rules

The rules of exponentiation are a set of guidelines for handling powers and roots in mathematical expressions. A fundamental rule is that of fractional exponents. The expression \(a^{n/m}\), for example, can be read as 'the mth root of \(a\) raised to the nth power' or 'the nth power of \(a\) rooted by m.'

In the exercise, the expression \(27^{2 / 3}\), we first raise 27 to the second power (27 squared is 729) and then we take the cube root of this result. Remember, with fractional exponents, if we have an expression \(a^{1/n}\), this is equivalent to the nth root of \(a\), and vice versa; knowing this is particularly helpful when expressing roots in different forms.