Problem 34 - Brief Applied Calculus

Question

Evaluate each expression without using a calculator. $$ 9^{-1 / 2} $$

Step-by-Step Solution

Answer

\(\frac{1}{3}\)

Step 1: Understand the Expression

The expression given is \(9^{-1/2}\). The exponent here is \(-1/2\), which can be interpreted as a negative fractional exponent.

Step 2: Convert the Negative Exponent

Recall that a negative exponent indicates the reciprocal of the base raised to the positive of that exponent. For example, \(a^{-b} = \frac{1}{a^b}\). Applying this to the expression \(9^{-1/2}\) gives \(\frac{1}{9^{1/2}}\).

Step 3: Simplify the Fractional Exponent

A fractional exponent \(a^{1/2}\) represents a square root. Hence, \(9^{1/2}\) is the same as \(\sqrt{9}\).

Step 4: Calculate the Square Root

\(\sqrt{9} = 3\), so we substitute this back into the expression to get \(\frac{1}{3}\).

Step 5: Write the Final Answer

The expression \(9^{-1/2}\) simplifies to \(\frac{1}{3}\).

Key Concepts

Negative Exponents

Let's understand negative exponents. In mathematics, an exponent indicates the number of times a base is multiplied by itself. For example, in \(3^2\), 3 is multiplied by itself twice to give \(9\).
However, when the exponent is negative, like in \(a^{-1}\), this signifies the reciprocal of the base raised to the positive exponent. Simply put, \(a^{-b} = \frac{1}{a^b}\).

• Instead of multiplication, think about division or taking the reciprocal.
• This is how we transform \(9^{-1/2}\) into \(\frac{1}{9^{1/2}}\).

Understanding this rule helps simplify expressions without direct calculation, underscoring an essential part of exponent knowledge.

Reciprocal

The concept of the reciprocal is quite straightforward once you break it down. A reciprocal of a number is essentially flipping the numerator and the denominator of a fraction.
For any non-zero number \(a\), the reciprocal is \(\frac{1}{a}\). For instance, the reciprocal of 5 is \(\frac{1}{5}\).

• When you encounter expressions like \(9^{-1/2}\), the first action is to find the reciprocal.
• This results in \(\frac{1}{9^{1/2}}\), simplifying our process by setting up for further calculations.

Reciprocals are vital when dealing with divisions and negative exponents. They are the stepping stone that paves the way to efficiently simplifying an expression.

Square Roots

Square roots play a crucial role when dealing with fractional exponents. A square root occurs when you find a number which, when multiplied by itself, gives the original number.
For example, the square root of 9 is 3 because \(3 \times 3 = 9\). In mathematical terms, it is expressed as \(\sqrt{9} = 3\) or \(9^{1/2} = 3\).

• Fractional exponents, like \(9^{1/2}\), are interpreted as square roots.
• This understanding makes it clear why \(\sqrt{9}\) is used in simplifying \(9^{-1/2}\).

Once you simplify the square root, you can proceed with the rest of the problem smoothly, as seen in reducing \(9^{1/2}\) to obtain \(\frac{1}{3}\) in the original expression.