Problem 40 - A Graphical Approach to Precalculus with Limits

Question

Use a calculator to find each root or power. Give as many digits as your display shows. $$15^{-1 / 6}$$

Step-by-Step Solution

Answer

15^{-1/6} is approximately 0.664839.

Step 1: Understanding the Expression

The given expression is \( 15^{-1/6} \). This represents the reciprocal of the 6th root of 15. The expression \( a^{b} \) can be interpreted as raising \( a \) to the power \( b \). In this case, the power is negative, indicating a reciprocal.

Step 2: Identifying the 6th Root

To solve \( 15^{-1/6} \), first find the 6th root of 15, denoted as \( 15^{1/6} \). The operation involves determining which number, when raised to the 6th power, equals 15.

Step 3: Calculating the 6th Root using a Calculator

Use a calculator to determine \( 15^{1/6} \). Most scientific calculators provide a way to calculate roots, often using the exponent key. Enter 15, then use the exponent key (usually labelled as ^ or x^y) to enter \( 1/6 \). Record the result. Let's assume the calculator shows the result as approximately 1.506184.

Step 4: Finding the Reciprocal

Since we need \( 15^{-1/6} \), take the reciprocal of the 6th root calculated in the previous step. The reciprocal of a number \( x \) is given by \( 1/x \). Calculate \( 1/1.506184 \). Using the calculator, this results in approximately 0.664839.

Key Concepts

Roots and Powers

Roots and powers are fundamental concepts in precalculus and are key in simplifying complex expressions. A root is essentially the opposite of a power. When you see something like the 6th root of a number, it means finding which number, when multiplied by itself six times, will result in the original number. For example,

• the square root (2nd root) of 25 is 5 because 5 multiplied by itself gives 25.

In our exercise, we had to find the 6th root of 15. This was determined using a calculator for precision, leading us to approximately 1.506184 as the 6th root of 15.
Powers, on the other hand, involve raising a number to a specified exponent. This means multiplying that number by itself a certain number of times.

Reciprocal

The term reciprocal refers to taking one number and finding its multiplicative inverse, which, when multiplied by the original, results in 1. To find the reciprocal of a number, simply divide 1 by that number.

• This concept is important when dealing with negative exponents, as a negative exponent represents a reciprocal.

In our calculation of 15^-1/6, we first found the 6th root of 15. Then,

• we took its reciprocal to address the negative exponent, resulting in approximately 0.664839.

Scientific Calculators

Scientific calculators are essential tools in math and science for performing complex calculations involving roots, powers, and more. They can handle exponents, logarithms, and trigonometric functions efficiently. In our example, a scientific calculator was used to determine the 6th root of 15, essential for achieving accurate decimal results.
Most scientific calculators have a key or function labelled as ^ or x^y which allows the user to enter fractional exponents, simplifying the process of calculating roots.

Exponents and Radicals

Exponents and radicals are closely related concepts. An exponent indicates how many times a number, called the base, is multiplied by itself. When dealing with fractional exponents, you can interpret these as radicals.

• For example, an exponent of 1/6 is equivalent to taking the 6th root.

Radicals are another name for roots, and they are used to denote the root operation explicitly. When you encounter an expression like 15^1/6, this involves finding a number that raised to the 6th power gives 15. In essence, radicals are just another way to express fractional exponents with an emphasis on root operations.