Problem 7 - Algebra 1: Concepts and Skills

Question

Evaluate the expression without using a calculator. $$ 9^{3 / 2} $$

Step-by-Step Solution

Answer

The evaluated expression is \(27\).

Step 1: Solve the denominator of the exponent first

A fraction exponent can be solved by dealing with the denominator first. The denominator of the fraction is \(2\). This means we need to take the square root of the base. In our expression, the base is \(9\). The square root of \(9\) is \(3\). So, \(9^{1 / 2}\) equals \(3\).

Step 2: Solve the numerator of the exponent

The numerator of the fraction is \(3\), which means we need to raise the result from step 1 to the power of \(3\). So, \(3^{3} = 3 * 3 * 3 = 27\).

Step 3: Final result

Therefore, the evaluation of \(9^{3 / 2}\) gives the result of \(27\).

Key Concepts

Fractional Exponents Demystified

Fractional exponents might look a bit intimidating at first, but they are actually a straightforward way of expressing roots alongside powers. Let's break down the fraction: in an expression like \(a^{m/n}\), the base \(a\) is handled by first focusing on the denominator \(n\) and then the numerator \(m\). This means you first take the \(n\)-th root of \(a\) and then raise the result to the \(m\)-th power.
Take for example \(9^{3/2}\):

• The denominator \(2\) guides us to first find the square root of \(9\), which is \(3\).
• Next, the numerator \(3\) tells us to cube this result, thus \(3^3 = 27\).

This two-step process of handling fractional exponents by separating it into a root and a power makes complex calculations much simpler.

Understanding the Square Root

The square root is a fundamental concept, often encountered in fractional exponents. But what does it really mean? The square root of a number is a value which, when multiplied by itself, gives the original number back. For instance, the square root of \(9\) is \(3\) because \(3 \times 3 = 9\).
In fractional exponent scenarios like \(9^{1/2}\), the denominator \(2\) signals that we're determining the square root.
This idea extends to other roots too, such as cube roots and beyond.
Understanding this concept is crucial, because it often serves as the foundation for solving more complex problems involving fractional exponents. It helps to visualize it as the reverse operation of squaring a number.

Unpacking Powers of Numbers

Raising a number to a power involves multiplying that number by itself a specified number of times. This is known as exponentiation. A simple example is \(3^3\), which calculates as \(3 \times 3 \times 3 = 27\).
Frequently, powers show up in expressions with fractional exponents, where you first deal with the root before tackling the power.

• In \(9^{3/2}\), once you've taken the square root to get \(3\), your next task is to raise \(3\) to the third power, yielding \(27\).

This concept of powers serves as a cornerstone in mathematics, underpinning everything from basic arithmetic to advanced calculus. Recognizing how to handle powers efficiently can greatly simplify the process of many mathematical operations.