Problem 61 - Algebra 1

Question

Use a calculator to evaluate $$6^{9}$$

Step-by-Step Solution

Answer

The result obtained by the calculator when evaluating \(6^{9}\) is 10077696.

Step 1: Understand the problem

The problem is asking to evaluate \(6^{9}\). This implies that we multiply 6 by itself 9 times.

Step 2: Multiply

We need to input the expression \(6^{9}\) into the calculator. The '^' symbol stands for 'raised to the power of'. When we input this into the calculator, it will mechanically perform the multiplication of 6 by itself 9 times.

Step 3: Output the result

After inputting the expression in the calculator, we get the result.

Key Concepts

Using a Calculator

When tasked with evaluating exponents like \(6^9\), a calculator becomes an invaluable tool. Modern calculators, including those on smartphones, scientific calculators, or dedicated calculator apps, generally have a specific key to handle exponentiation. This key might look like a caret (^), as the exponentiation process involves raising a number to a specific power. To use a calculator for this task:

• First, identify and turn on your calculator. Ensure that it supports exponentiation by finding the appropriate button.
• Input the base number first. In the case of \(6^9\), type 6 into the calculator.
• Next, find the exponentiation button — often marked as '^', 'EXP', or 'y^x'. Use this button to indicate that you want to raise the number to a power.
• Finally, input the exponent number, which is 9 in this scenario, and press the equal sign if needed to get your result.

This method quickly gives you the answer, eliminating human calculation errors.

Exponent Notation

Exponent notation is a shorthand used in mathematics to express repeated multiplication of the same number. When you see a number written in the form \(a^n\), it means 'a' is multiplied by itself 'n' times. This type of notation simplifies the expression of very large numbers and makes calculations more manageable.

• \(a\) is called the base—it is the number you are repeatedly multiplying.
• \(n\) is the exponent—it tells you how many times to multiply the base by itself.

For example:- In \(6^2\), the base is 6, and it is multiplied by itself twice, equaling 36.- In \(6^9\), as in the exercise, you multiply 6 by itself 9 times.Exponents help in describing phenomena such as growth rates and scientific measurements efficiently.

Powers of Numbers

The concept of powers of numbers expands upon simple multiplication by creatively using exponents to signify how many times a base number is used in a product. This mathematical tool is essential in various fields, from physics to finance, due to its ability to easily convey very large numbers.Let's break it down:- When a number \(x\) is raised to the power of 2, it is often referred to as 'squared'. This means \(x\times x\). - When raised to the power of 3, it is 'cubed', representing \(x \times x \times x\).- Higher powers, like \(x^9\), represent the base number multiplied by itself 9 times, a scenario increasingly best handled by using calculators or software.This notation is not merely about simplification. It opens up pathways for solving equations, understanding exponential growth or decay, and translating real-world scenarios into mathematical expressions. Understanding powers of numbers gives you a powerful toolset for tackling both theoretical and practical problems.