Problem 34 - Applied Calculus

Question

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

Step-by-Step Solution

Answer

\(9^{-1/2} = \frac{1}{3}\).

Step 1: Understanding the Problem

The expression given is \(9^{-1/2}\). We need to understand what each part of the expression represents. The base is 9, the exponent is \(-1/2\). The exponent \(-1/2\) includes a negative sign, which indicates a reciprocal, and a fractional part, which indicates a root. Hence, we need to find the reciprocal of the square root of 9.

Step 2: Finding the Square Root

First, we focus on the fractional part of the exponent. The fraction \(1/2\) indicates a square root operation. We calculate the square root of 9. Since 9 is a perfect square, its square root is 3.

Step 3: Applying the Negative Exponent

Now, we address the negative sign in the exponent \(-1/2\). A negative exponent means we take the reciprocal of the base raised to the corresponding positive exponent. The reciprocal of the square root of 9, which is 3, becomes \(1/3\). Thus, we replace \(9^{-1/2}\) with \(1/3\).

Step 4: Final Evaluation

Putting it all together, \(9^{-1/2} = \frac{1}{3}\). Therefore, the expression evaluates to \(\frac{1}{3}\).

Key Concepts

Negative Exponents

When dealing with exponents, encountering negative values might seem tricky at first. However, understanding negative exponents is quite simple! A negative exponent indicates that we take the reciprocal of the base raised to the positive version of the exponent.
For example:

• If we have something like \( 9^{-1} \), it means \( \frac{1}{9} \).
• For any base \( a \), \( a^{-n} \) equals \( \frac{1}{a^n} \).

The negative sign helps us "flip" the base from the numerator to the denominator or vice versa. It’s a way of representing how many times you’d divide by the base instead of multiplying. So remember, negative exponents are just another way to express fractions.

Fractional Exponents

Fractional exponents can look confusing initially, but they're just another form of expressing roots. When you see a fraction as an exponent, it signifies a power and a root.
Take for instance \( a^{1/2} \). The exponent \( 1/2 \) represents the square root of \( a \).
Essentials to remember about fractional exponents:

• The numerator of the fraction exerts a power over the base.
• The denominator indicates the root we must take.

For example:

• In \( 27^{1/3} \), the denominator tells us to take the cube root of 27.
• For \( 81^{3/4} \), you take the fourth root of 81 and cube the result.

Understanding these exponents helps simplify problems involving roots and powers quickly.

Square Roots

Square roots are fundamental in math, serving as the opposite of squaring a number. When you square a number, you multiply it by itself.
Likewise, finding a square root means identifying a number that, when squared, gives you the original number.
Basics of square roots:

• The square root of 9 is 3 because \( 3 \times 3 = 9 \).
• The notation for square root is \( \sqrt{} \).

Some numbers, like 9 or 16, are perfect squares, making their roots whole numbers.
Numbers that aren't perfect squares result in roots that are decimals or fractions. This idea connects with fractional exponents, where \( a^{1/2} \) means the square root of \( a \).
Understanding this can be very useful, especially when you're simplifying complex expressions!