Problem 80 - Fundamentals of Mathematics

Question

Use a calculator with the keys \(\sqrt{x}, y^{x}\), and \(1 / x\) to find each of the values. \(\sqrt{46,225}\)

Step-by-Step Solution

Answer

The square root of 46,225 is 215.

Step 1: Understand the problem

We need to find the square root of the number 46,225 using a calculator with specific keys.

Step 2: Identify the available keys

The calculator keys available are: square root (\( \sqrt{x} \)), exponentiation (\( y^x \)), and reciprocal (\( 1/x \)). For this problem, we will specifically use the square root key (\( \sqrt{x} \)).

Step 3: Enter the number into the calculator

Enter the number 46,225 into the calculator. This is the value for which we want to find the square root.

Step 4: Apply the square root operation

Press the square root key (\( \sqrt{x} \)) on the calculator. This will calculate the square root of the number entered in the previous step.

Step 5: Record the result

The calculator should display the result as 215, which is the square root of 46,225.

Key Concepts

Square Root

The square root of a number is a value that, when multiplied by itself, gives the original number. In this case, the square root of 46,225 is calculated. The square root symbol is denoted as \( \sqrt{x} \). Finding the square root can simplify complex numbers by breaking them down. To use a calculator for this, enter the number and press the \( \sqrt{x} \) key. For example, entering 46,225 and pressing \( \sqrt{x} \) will give you 215. This means \( 215 \times 215 = 46,225 \). Understanding square roots:

• It reverses the process of squaring a number.
• Every positive number has a positive square root.
• Perfect squares result in whole numbers.

Remember, the square root operation gives the principal (positive) root. Always double-check your results with estimation to see if they make sense.

Exponentiation

Exponentiation involves raising a number, the base, to the power of an exponent. It is expressed as \( y^x \), meaning "\( y \) raised to the \( x \)". This mathematical operation is fundamental for working with powers. To use exponentiation with a calculator, enter the base first, press the exponentiation key, and then enter the exponent. How exponentiation works:

• Exponent indicates how many times the base multiplies by itself.
• A number with exponent 1 is the number itself \( (y^1 = y) \).
• An exponent of 0 results in 1 \( (y^0 = 1) \), for non-zero \( y \).

This operation is used for much more than just multiplication. It helps simplify numbers and solve equations efficiently, especially with larger bases and exponents.

Reciprocal Calculation

The reciprocal of a number is 1 divided by that number, denoted as \( \frac{1}{x} \). It is useful for division problems where you invert multiplication. To calculate the reciprocal, you can use a calculator's reciprocal key \( 1/x \). For instance, if you input a number and press \( 1/x \), you access its reciprocal result. Key insights into reciprocals:

• Multiplying a number by its reciprocal always gives 1.
• The reciprocal of a fraction flips the numerator and denominator.
• Zero does not have a reciprocal due to division by zero.

Understanding reciprocals strengthens your capacity for more complex mathematical operations, particularly in algebra and calculus. By mastering reciprocals, you improve your problem-solving tools and tactics.