Problem 29 - Algebra 2

Question

Evaluate each expression. $$ 81^{-\frac{1}{2}} \cdot 81^{\frac{3}{2}} $$

Step-by-Step Solution

Answer

The expression evaluates to 81.

Step 1: Understanding Negative Exponents

Negative exponents indicate that you take the reciprocal of the base. Therefore, \( 81^{-\frac{1}{2}} \) is equivalent to \( \frac{1}{81^{\frac{1}{2}}} \), which further simplifies to \( \frac{1}{\sqrt{81}} \). Since \( \sqrt{81} = 9 \), this becomes \( \frac{1}{9} \).

Step 2: Simplifying the Original Expression

The expression \( 81^{-\frac{1}{2}} \cdot 81^{\frac{3}{2}} \) can be simplified using the property of exponents that states \( a^m \cdot a^n = a^{m+n} \). Therefore, \( 81^{-\frac{1}{2}} \cdot 81^{\frac{3}{2}} = 81^{(-\frac{1}{2})+(\frac{3}{2})} \).

Step 3: Adding the Exponents

Add the exponents \(-\frac{1}{2}\) and \(\frac{3}{2}\). The result is \[ -\frac{1}{2} + \frac{3}{2} = \frac{2}{2} = 1 \]. This means that \( 81^{(-\frac{1}{2})+(\frac{3}{2})} = 81^1 \).

Step 4: Evaluating the Final Expression

The expression simplifies to \( 81^1 \), which is just 81. Therefore, \( 81^{-\frac{1}{2}} \cdot 81^{\frac{3}{2}} \) simplifies to 81.

Key Concepts

Negative Exponents

Negative exponents can be a bit tricky at first, but they hold a simple rule: they mean you should take the reciprocal of the base. For example, when you see a negative exponent like in \( 81^{-\frac{1}{2}} \), it's telling you to flip the base from the numerator to the denominator, or vice versa. This means that \( 81^{-\frac{1}{2}} \) becomes \( \frac{1}{81^{\frac{1}{2}}} \).
Once it's in this form, you can evaluate the expression further by finding the square root of 81. So, \( \sqrt{81} = 9 \), which simplifies the expression to \( \frac{1}{9} \). This idea of reciprocal is key to understanding negative exponents. Keep this logic in mind whenever you encounter them!

Exponent Laws

Exponent laws are essential rules that help simplify expressions efficiently. They provide a framework for handling expressions involving powers. One of the most used laws is the multiplication of exponents: if you have the same base, you can add the exponents.

• For example: \( a^m \cdot a^n = a^{m+n} \).

In the expression \( 81^{-rac{1}{2}} \cdot 81^{rac{3}{2}} \), both terms have the base 81. Thus, the rule allows us to add the exponents together:
\[ 81^{(-\frac{1}{2})+(\frac{3}{2})} \].
This simplifies the expression into one single exponent, making it much easier to evaluate. Being familiar with these laws means you'll be able to tackle exponentiation problems more confidently.

Simplifying Expressions

Simplifying an expression involves breaking it down to its most basic form without changing its value. This process often uses exponent laws and basic arithmetic. For the expression \( 81^{-rac{1}{2}} \cdot 81^{rac{3}{2}} \), simplifying involves first using the exponent law to add the exponents:
\[ -\frac{1}{2} + \frac{3}{2} = \frac{2}{2} = 1 \].
This results in \( 81^1 \), which is simply 81.By simplifying expressions, you're not only making them easier to work with but also ensuring they are in a form that is straightforward to interpret. This is often an important step in problem-solving as it reduces complicated problems into something manageable.

Square Roots

The square root is a fundamental concept that appears frequently in mathematics. It is the number that, when multiplied by itself, gives the original number. For example, the square root of 81 is 9, because \( 9 \times 9 = 81 \).
When you are dealing with expressions like \( 81^{\frac{1}{2}} \), you are essentially looking for the number that squares back to the base, which means \( 81^{\frac{1}{2}} \) evaluates to \( \sqrt{81} \). Understanding how square roots interact with exponents can simplify calculations. They frequently show up in problems involving negative exponents and fractional exponents, where they help transform complex expressions into more manageable numbers.