Question

Evaluate the expression using a calculator. Round your answer to two decimal places when appropriate. (See Example 3.) \(25^{-1 / 3}\)

Short answer

The result of evaluating the expression \(25^{-1 / 3}\) and rounding it to two decimal places is 0.34.

Step-by-step solution

Understand the expression

The expression \(25^{-1 / 3}\) can be expressed as \(\frac{1}{\sqrt[3]{25}}\). This is because a negative exponent means taking the reciprocal of the base, and the fraction in the exponent signifies a root. In this case, -1 means we take the reciprocal and 3 in the denominator of the exponent means we take the cube root.

Evaluate the cube root

Start by evaluating the cube root of 25 which is approximately 2.92402.

Take the reciprocal

After getting the cube root, you then calculate its reciprocal. That gives you 1 divided by 2.92402 which equals approximately 0.342.

Round to two decimal places

Lastly, round off the result from step 3 to two decimal places. The answer will then become 0.34.

Key concepts

Negative Exponents

When dealing with negative exponents like in the expression \(25^{-1/3}\), it's essential first to understand what they represent. A negative exponent tells us to take the reciprocal of the base number. For example, \(a^{-n}\) is equivalent to \(\frac{1}{a^n}\). The concept can seem confusing, but it essentially means that you're dividing 1 by the base number raised to the positive exponent.

Thinking about the reciprocal helps us simplify the expression before we proceed with additional steps. For instance, in \(25^{-1/3}\), we first recognize that the reciprocal of 25 is \(\frac{1}{25}\), and then we consider the cube root, which is indicated by the fraction \(1/3\).

Cube Root

The cube root, symbolized by \(\sqrt[3]{x}\), is a number that, when multiplied by itself three times, gives the original number x. In our expression, calculating the cube root of 25 helps us find the number which we can cube (multiply by itself, three times) to get 25. While the cube root of a perfect cube, such as 8 or 27, can be found exactly, other numbers like 25 require approximation.

To find \(\sqrt[3]{25}\) without a calculator, we could guess and check by finding a number close to \(\sqrt[3]{27}\), which is 3. Given that 25 is a little less than 27, we could estimate that the cube root of 25 is a little less than 3. A calculator refines this guess to a more precise decimal.

Reciprocal of a Number

The reciprocal of a number is what you multiply it by to get the value 1. It's like flipping a fraction upside down. The reciprocal of a whole number is 1 divided by that number. For example, the reciprocal of 5 is \(\frac{1}{5}\). Similarly, for a fraction such as \(\frac{x}{y}\), the reciprocal is \(\frac{y}{x}\). In the context of our original problem, after finding the approximate cube root of 25 to be 2.92402, we then find its reciprocal by calculating \(\frac{1}{2.92402}\), which gives us another decimal that we'll soon round off.

Rounding Decimals

Rounding decimals is a way to simplify numbers, making them easier to work with. To round a number to two decimal places, you look at the third decimal place. If the third decimal place is 5 or higher, you round up. If it's lower than 5, you keep the second decimal place as is.

In the solution for \(25^{-1/3}\), the third step resulted in a long decimal number. When we round 0.342 to two decimal places, we look at the second decimal place (4), and then right after it (the third decimal place), which is 2. Since 2 is less than 5, we keep the second decimal place as 4, resulting in the rounded number 0.34.