Problem 66 - Intermediate Algebra

Question

Simplify each expression. All variables represent positive real numbers. See Example 7. $$ 49^{-1 / 2} $$

Step-by-Step Solution

Answer

The simplified expression is \(\frac{1}{7}\).

Step 1: Understand the Meaning of the Exponent

The expression contains an exponent of \(-1/2\). The exponent \(-1\) indicates the reciprocal of the base, and \(1/2\) implies taking the square root. So, \(49^{-1/2}\) indicates the reciprocal of the square root of 49.

Step 2: Calculate the Square Root

Identify the square root of 49. The square root of 49 is 7 since \(7 \times 7 = 49\). Thus, \(\sqrt{49} = 7\).

Step 3: Find the Reciprocal

The exponent \(-1\) means that we need to take the reciprocal of the result from the previous step. The reciprocal of 7 is \(\frac{1}{7}\).

Step 4: Combine the Steps

Combine the findings from the previous steps, resulting in the simplified expression. This means \(49^{-1/2} = \frac{1}{7}\).

Key Concepts

Negative Exponents

When you see a negative exponent, it might look a bit intimidating at first, but don't worry. It's not as complex as it seems. A negative exponent simply tells you to take the reciprocal of the base. The reciprocal of a number is just 1 divided by that number.
For instance:

• For the expression "\(3^{-1}\)", you can rewrite it as "\(\frac{1}{3}\)".
• If you see "\(x^{-2}\)", you can think of it as "\(\frac{1}{x^2}\)".

It's really about flipping the base's position from the numerator to the denominator or vice versa depending on where it was originally.
This helps simplify expressions with negative exponents, making them easier to work with.

Square Roots

Square roots are another essential concept, especially when simplifying expressions like in our problem. The square root of a number is the value that, when multiplied by itself, gives the original number. It's denoted by the radical symbol \(\sqrt{}\).
Here's how it works:

• The square root of 4 is 2 because \(2 \times 2 = 4\).
• Similarly, the square root of 9 is 3 since \(3 \times 3 = 9\).

It's important to remember that real numbers under a square root must be non-negative because we're dealing with real numbers. When simplifying expressions, finding the square root is often the first step before applying other operations.

Reciprocal

The concept of a reciprocal is quite straightforward. To find the reciprocal of a number, you simply flip it—if it's in the form of a fraction \(\frac{a}{b}\), the reciprocal is \(\frac{b}{a}\). If the number is a whole number or an integer, imagine it as a fraction over 1.

• For instance, the reciprocal of 5 is \(\frac{1}{5}\).
• Similarly, the reciprocal of \(\frac{3}{4}\) is \(\frac{4}{3}\).

When dealing with negative exponents, you typically end up calculating a reciprocal as a part of simplifying your expression. This concept is fundamental because it allows you to convert expressions into their inverse forms, making computations and simplifications more manageable.