Problem 5 - A Graphical Approach to College Algebra

Question

Evaluate each expression. Do not use a calculator. $$81^{3 / 2}$$

Step-by-Step Solution

Answer

The expression evaluates to 729.

Step 1: Understand the Expression

The expression we have is \(81^{3/2}\). This is a power with a rational (fraction) exponent, which means we can use the laws of exponents to simplify it.

Step 2: Break Down the Exponent

The exponent \( \frac{3}{2} \) implies two operations: a square root (denominator) and a power (numerator). Specifically, \(81^{3/2} = (81^{1/2})^3\). This means we first take the square root of 81, then raise it to the power of 3.

Step 3: Calculate the Square Root

Calculate \(81^{1/2}\), which is the square root of 81. Since 81 is a perfect square, applying the square root operation gives us 9 because \(9 \times 9 = 81\).

Step 4: Raise to the Power of 3

Raise the result from the previous step to the power of 3: \(9^3\). We need to calculate this by performing \(9 \times 9 \times 9\). First, calculate \(9 \times 9 = 81\), then multiply 81 by 9, giving \(81 \times 9 = 729\).

Step 5: Final Result

The evaluation of the expression \(81^{3/2} = 729\). We have completed the calculation by breaking it down into a square root and then raising the result to the third power.

Key Concepts

Exponent Laws

Exponent laws help us understand how to handle expressions like \(81^{3/2}\) with great ease. When dealing with rational exponents, a key law to remember is that \(a^{m/n}\) can be interpreted in two main ways:

• As \((a^{1/n})^m\), which means you take the \(n\)-th root of \(a\) first, then raise the result to the \(m\)-th power.
• Alternatively, as \((a^m)^{1/n}\), where you raise \(a\) to the \(m\)-th power first, and then find its \(n\)-th root.

Both methods will yield the same result. This characteristic of rational exponents is crucial as it provides flexibility in solving expressions and finding solutions. In our example, we chose \((81^{1/2})^3\) because it is often easier to find the square root (\(1/2\)-th power) before raising to the cube (\(3\)-rd power). Choosing the right approach can simplify your calculations significantly.
Exponent laws not only help us rewrite and evaluate expressions with fractional exponents, but are also essential for understanding broader concepts in algebra and calculus.

Square Roots

Square roots play a vital role when working with rational exponents, as we see in \(81^{3/2}\). The square root of a number is what you multiply by itself to get that number. For instance, we take \(81\) and look for a number which, when squared, equals \(81\). This number is \(9\), since \(9 \times 9 = 81\).
In mathematics, taking the square root is often expressed as raising the number to the \(1/2\) power, so \(81^{1/2} = 9\). This is a straightforward application of the exponent law \(a^{m/n}\) we discussed earlier.
It's helpful to recognize common perfect squares, such as \(4, 9, 16, 25,\) and so on, as these numbers make calculating square roots without a calculator much easier. This is because the square root of each perfect square is an integer, simplifying the process dramatically.

Perfect Squares

Understanding perfect squares will make evaluating expressions involving square roots much smoother. A perfect square is a number that has a whole number as its square root. For example, \(81\) is a perfect square because \(9 imes 9 = 81\), and \(9\) is a whole number.
Memorizing common perfect squares can greatly speed up your math work. Here are a few examples:

• \(1^2 = 1\)
• \(2^2 = 4\)
• \(3^2 = 9\)
• \(4^2 = 16\)
• \(5^2 = 25\)
• \(6^2 = 36\)
• \(7^2 = 49\)
• \(8^2 = 64\)
• \(9^2 = 81\)

This list helps in quick identification of which numbers are perfect squares. Knowing these can make it much quicker to evaluate expressions like \(81^{3/2}\). By recognizing \(81\) as a perfect square, you simplify the step of finding its square root, moving quickly towards your final answer.