Problem 89 - Fundamentals of Mathematics

Question

Use a calculator with the keys \(\sqrt{x}, y^{x}\), and \(1 / x\) to find each of the values. \(\sqrt[3]{531,441}\)

Step-by-Step Solution

Answer

The cube root of 531,441 is 81.

Step 1: Rewrite the Expression using Exponents

The expression \( \sqrt[3]{531,441} \) can be rewritten using exponents as \( 531,441^{1/3} \). Here we need to find the cube root of 531,441.

Step 2: Use the Exponent Key

To calculate the cube root, input 531,441 and then use the exponent key \( y^{x} \). Enter the fraction 1/3 as the exponent. This is executed as: \( 531,441^{1/3} \).

Step 3: Calculate the Result

The calculator processes the input, and the displayed result will be \( 81 \), since \( 81^3 = 531,441 \).

Key Concepts

Exponents

Exponents are a fundamental concept in mathematics, serving as a shorthand way to represent repeated multiplication of a number by itself. For example, in the expression \( 2^5 \), the number 2 is the base, and 5 is the exponent. This tells us that we need to multiply 2 by itself five times:

• \( 2 \times 2 \times 2 \times 2 \times 2 = 32 \)

Exponents can also be fractions, allowing us to express roots as exponentiations. When we see \( x^{1/3} \), it represents the cube root of \( x \). This means we need to find a number which, when multiplied by itself three times, gives \( x \). In our example, \( 531,441^{1/3} \) evaluates to 81 because \( 81 \times 81 \times 81 = 531,441 \). Understanding how to manipulate exponents is crucial for solving various mathematical problems, from basic algebra to more advanced calculus.

Cube Root

The cube root of a number is the value that, when used in threefold multiplication, returns the original number. It is similar to the square root, but instead of finding a number which squares to the original number, we find one that "cubes" to it. A cube root is generally symbolized as \( \sqrt[3]{x} \) or using exponents as \( x^{1/3} \). For example, the cube root of 8 is 2 because multiplying 2 by itself twice (i.e., \( 2 \times 2 \times 2 \)) results in 8. When dealing with large numbers, like in our exercise with 531,441, calculating cube roots manually can be time-consuming and complex. However, modern calculators simplify this process using specific functions that allow quick and accurate computation, thus removing potential manual calculation errors.

Calculator Functions

Modern calculators come equipped with various functions that make complex mathematical operations more accessible. Three functions particularly useful for solving problems involving exponents and roots are the square root \( \sqrt{x} \), power \( y^x \), and reciprocal \( 1/x \) functions.

• Square Root (\( \sqrt{x} \)): Calculates the square root of a number.
• Power (\( y^x \)): Allows any number to be raised to a specific power. For instance, entering 4 for \( y \) and 3 for \( x \) gives 64 since \( 4^3 = 64 \).
• Reciprocal (\( 1/x \)): Finds the reciprocal or the multiplicative inverse of a number.

To solve the exercise given, the power function is particularly highlighted. By inputting 531,441 and using \( y^x \) with \( x \) as \( 1/3 \), the calculator accurately finds the cube root, providing a precise result of 81 efficiently. Using these calculator functions helps students solve exponents and roots problems quickly, enhancing learning and understanding.