Question

Evaluate the expression using a calculator. Round your answer to two decimal places when appropriate. (See Example 3.) \(\sqrt[7]{1695}\)

Short answer

The 7th root of 1695, rounded to two decimal places, is approximately 3.64.

Step-by-step solution

Understanding the question

The exercise wants to find the 7th root of 1695. This can be written in mathematical terms as \(\sqrt[7]{1695}\)

Use a calculator

Use a scientific calculator to find this value. Most calculators have a function for calculating roots. Input 7 as the root and 1695 as the number.

Round to the appropriate decimal

Once the calculator gives a value, it should be rounded to two decimal places. Remember that if the third digit after the decimal point is 5 or more, round up the second digit by one.

Key concepts

Scientific Calculator

A scientific calculator is an essential tool for solving complex mathematical problems accurately and efficiently. Unlike basic calculators, which primarily handle arithmetic operations, scientific calculators come equipped with a range of functions including logarithms, roots, trigonometry, and exponents. When calculating the 7th root of a number such as 1695, a scientific calculator can simplify this task greatly.
To use a scientific calculator for finding roots, you typically look for a key that is dedicated to calculating roots or use the 'y^x' function, setting 1/7 as the power when solving \(x^{1/7}\). If you're unsure of the exact process on your specific model, consulting the calculator's manual can offer guidance specific to your device.
This capability of scientific calculators not only saves time but also reduces the risk of computational errors, allowing a student to confidently explore more advanced mathematical concepts without the distraction of manual calculation errors.

Rounding Decimals

Rounding decimals is a useful mathematical technique that ensures numerical results are presented in a simplified form that's easier to communicate and understand, especially in everyday situations. In our exercise, after determining the 7th root of 1695 using a scientific calculator, the resulting decimal number needs to be rounded to two decimal places.
Here's how rounding works: Look at the third decimal place. If this number is 5 or greater, increase the second decimal place by one. If it's less than 5, do not change the second decimal place. For instance, if your result is 3.478, you would round it to 3.48.

• Numbers such as 3.456 would round to 3.46 because the third digit (6) is greater than 5.
• Numbers like 3.452 would round to 3.45 since the third digit (2) is less than 5.

This standard practice of rounding not only makes numbers more manageable but also aligns them with common numerical standards used in scientific and statistical applications.

Mathematical Expressions

Mathematical expressions are combinations of numbers, variables, and operations (such as addition, subtraction, multiplication, division, and roots) that represent a particular value or set of values. In the context of this exercise, the expression \(\sqrt[7]{1695}\) is used to find the 7th root of 1695. This expression involves both a number and a root operation, indicating an advanced level of mathematics beyond basic arithmetic.
A firm understanding of how to interpret and manipulate mathematical expressions is crucial for success in math, science, and engineering fields. For this type of expression, breaking it down into manageable parts can simplify your approach. Firstly, recognize the base number (1695) and the root order (7), indicating that you're solving for a value that, when raised to the power of 7, equals 1695.
Understanding the language and logic of mathematical expressions empowers students to solve problems efficiently and apply these skills to more complex scenarios, solidifying their overall mathematical fluency.