Problem 10 - Intermediate Algebra

Question

For Problems \(1-30\), evaluate each numerical expression. $$ 81^{-\frac{1}{2}} $$

Step-by-Step Solution

Answer

The value of the expression \(81^{-\frac{1}{2}}\) is \(\frac{1}{9}\).

Step 1: Understand the expression

The problem asks us to evaluate the expression \(81^{-\frac{1}{2}}\). This involves a negative exponent, as well as a fractional exponent.

Step 2: Convert the negative exponent

A negative exponent indicates that we need to take the reciprocal of the base. So, \(81^{-\frac{1}{2}}\) can be rewritten as \(\frac{1}{81^{\frac{1}{2}}}\).

Step 3: Simplify the fractional exponent

The expression \(81^{\frac{1}{2}}\) represents the square root of 81. Therefore, we calculate \(\sqrt{81}\), which equals 9.

Step 4: Solve the expression

Substitute the value from Step 3 into the reciprocal. So, \(\frac{1}{81^{\frac{1}{2}}} = \frac{1}{9}\).

Step 5: Write the final answer

Hence, the value of the expression \(81^{-\frac{1}{2}}\) is \(\frac{1}{9}\).

Key Concepts

Negative Exponents

Negative exponents can initially be confusing, but they are quite straightforward once understood properly. When you encounter a negative exponent, such as in the expression \(81^{-\frac{1}{2}}\), it tells you to take the reciprocal of the base raised to the corresponding positive exponent. In simple terms, \(a^{-n} = \frac{1}{a^n}\).
For example, in the case of \(81^{-\frac{1}{2}}\), the negative exponent indicates you should write it as \(\frac{1}{81^{\frac{1}{2}}}\). This principle can be applied universally to any base with a negative exponent. Just remember:

• Change the sign of the exponent from negative to positive.
• Take the reciprocal of the base raised to the positive exponent.
• Follow through with further simplifications or calculations.

Understanding this conversion is critical to handling expressions involving negative exponents correctly.

Fractional Exponents

Fractional exponents represent roots of numbers, combining the idea of these two operations (exponentiation and rooting) into one simple, compact formula. Using the fractional exponent power, such as \(81^{\frac{1}{2}}\), it is equivalent to finding the square root of the base number.
Here's how it breaks down:

• The numerator (top number) of the fraction represents the power to which the base is raised.
• The denominator (bottom number) indicates the root you need to take of the base.

In the exercise example \(81^{\frac{1}{2}}\), the denominator is 2, which implies you take the square root of 81, resulting in 9. This method applies to all fractional exponents, for example, \(a^{m/n} = \sqrt[n]{a^m}\).
Practicing this will make it easier to interpret and solve these efficiently.

Square Roots

Square roots are a fundamental concept often paired with fractional exponents, as we've seen. Finding the square root means determining which number, when multiplied by itself, gives the original number. The square root of 81, often shown as \(\sqrt{81}\), equals 9.
This is because \(9 \times 9 = 81\).

• The square root (√) is a specific kind of root, indicated by the fractional exponent \(\frac{1}{2}\).
• Being able to recognize perfect squares helps in quickly solving root problems, like \(1, 4, 9, 16, 25, 36\), and so on.

Grasping the concept of square roots is crucial when dealing with more complex expressions, especially when they combine with other operations such as involving exponents.
Whether for purely numerical operations or algebraic expressions, the ability to efficiently compute square roots is a powerful tool in mathematics.