Problem 3 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Evaluate the expression. a. \(9(9)^{-1 / 2}\) b. \(5(5)^{-1 / 2}\)

Step-by-Step Solution

Answer

The given expressions can be evaluated as follows: a. \(9(9)^{-1/2} = 3\) b. \(5(5)^{-1/2} \approx 2.23\)

Step 1: Understand the rational exponent

The exponent -1/2 can be written as a combination of two exponents: -1 and 1/2. When an exponent is -1, it represents taking the reciprocal (i.e., flipping) of the number. When the exponent is 1/2, it represents taking the square root of the number. So, to evaluate the expression \(9(9)^{-1/2}\), we will first find the square root and then find the reciprocal of the result.

Step 2: Apply the exponent rules

We start by taking the square root of 9 and then finding the reciprocal of the result. Square root of 9: \(\sqrt{9}=3\) Reciprocal of the result: \(\frac{1}{3}\) Now we multiply 9 by the reciprocal of the result: \(9(\frac{1}{3}) = 3\) The expression a. \(9(9)^{-1/2}\) simplifies to 3. For expression b. \(5(5)^{-1/2}\):

Step 1: Understand the rational exponent

As explained before, the exponent -1/2 can be written as a combination of two exponents: -1 and 1/2. When an exponent is -1, it represents taking the reciprocal (i.e., flipping) of the number. When the exponent is 1/2, it represents taking the square root of the number. So, to evaluate the expression \(5(5)^{-1/2}\), we will first find the square root and then find the reciprocal of the result.

Step 2: Apply the exponent rules

We start by taking the square root of 5 and then finding the reciprocal of the result. Square root of 5 (rounded to two decimal places): \(\sqrt{5} \approx 2.24\) Reciprocal of the result: \(\frac{1}{2.24} \approx 0.446\) Now we multiply 5 by the reciprocal of the result: \(5(0.446) \approx 2.23\) The expression b. \(5(5)^{-1/2}\) simplifies to approximately 2.23.

Key Concepts

Reciprocal

The reciprocal of a number is basically its flipped version. To find the reciprocal of a number, you swap the numerator with the denominator. If the number is a whole number, you can think of it as being over 1 before flipping it upside down. For example, the reciprocal of 3 is \( \frac{1}{3} \). This concept is handy when dealing with negative exponents.

• When you see \( a^{-1} \), it is the same as \( \frac{1}{a} \).
• In the exercise, the exponent \(-1/2\) means you'll first find the reciprocal after taking the square root.

Understanding reciprocals can help simplify calculations involved in algebra, fractions, and when dealing with rational exponents.

Square Root

A square root of a number is essentially asking "what number multiplied by itself gives me this number?". Square roots are often represented by the radical symbol \( \sqrt{} \). For example, the square root of 9 is 3, since \(3 \times 3 = 9\).

• In the expressions \(9^{-1/2}\) and \(5^{-1/2}\), the \( \frac{1}{2} \) exponent signifies that we need to find the square root first.
• This is crucial because it transforms the original number into one that can be easily flipped as a reciprocal.

Once the square root is found, further calculations become simpler, especially when combined with exponent rules.

Exponent Rules

Exponent rules are guidelines that make handling exponents manageable, particularly with rational exponents. Some key rules include:

• Product of Powers: \( a^m \times a^n = a^{m+n} \)
• Power of a Power: \( (a^m)^n = a^{m \times n} \)
• Negative Exponent: \( a^{-m} = \frac{1}{a^m} \)
• Rational Exponent: \( a^{\frac{m}{n}} = \sqrt[n]{a^m} \)

For rational exponents, the fraction describes two operations: the numerator is the power, and the denominator is the root. For instance, in the exercises, \(-1/2\) suggests performing a square root operation followed by taking the reciprocal of the result.Understanding and applying these rules can simplify complex expressions, making them more approachable and easier to solve.