Problem 53 - Fundamentals of Mathematics

Question

Determine the value of each of the powers. Use a calculator to check each result. \(6^{9}\)

Step-by-Step Solution

Answer

The value of \(6^9\) is 10077696.

Step 1: Understanding the Base and Exponent

In the expression \(6^9\), 6 is the base number, and 9 is the exponent. This means we need to multiply the base, 6, by itself 9 times.

Step 2: Perform Repeated Multiplication

To find \(6^9\), multiply 6 by itself 9 times:\[ 6 \times 6 \times 6 \times 6 \times 6 \times 6 \times 6 \times 6 \times 6 \]}

Step 3: Calculate Step-by-Step

When calculated step by step: \(6 \times 6 = 36\)\(36 \times 6 = 216\)\(216 \times 6 = 1296\)\(1296 \times 6 = 7776\)\(7776 \times 6 = 46656\)\(46656 \times 6 = 279936\)\(279936 \times 6 = 1679616\)\(1679616 \times 6 = 10077696\)

Step 4: Check with a Calculator

Using a calculator to compute \(6^9\): Enter the number 6 and use the exponentiation function followed by 9 to get the result, confirming it as 10077696.

Key Concepts

Base Number

In mathematics, the concept of a base number is fundamental when working with exponents. The base number in an exponential expression dictates what number is being multiplied repeatedly. For example, in the expression \(6^9\), the base number is 6. This is the number that will be repeatedly multiplied by itself a specific number of times, depending on what the exponent indicates.

Understanding base numbers is important because they set the stage for calculating exponents accurately. Knowing the base helps you grasp the magnitude of the computation you'll be undertaking. Essentially, it's your starting point every time you're dealing with powers. In simple terms, think of the base number as your 'repeated actor,' and the repeated multiplication is the 'action' performed by the actor multiple times.

Repeated Multiplication

Repeated multiplication is at the heart of understanding how exponents work. When you see a number raised to a power, such as \(6^9\), you're being asked to multiply the base number by itself a certain number of times indicated by the exponent.

So, for \(6^9\), this means multiplying 6 by itself 9 times. This might initially seem daunting, but it can be broken down into manageable steps:

• Start by multiplying the base with itself (\(6 \times 6 = 36\)).
• Continue multiplying the result by the base (\(36 \times 6 = 216\)), and so on.

By breaking down the steps, you get:

• \(6 \times 6 = 36\)
• \(36 \times 6 = 216\)
• \(216 \times 6 = 1296\)
• \(1296 \times 6 = 7776\)
• \(7776 \times 6 = 46656\)
• \(46656 \times 6 = 279936\)
• \(279936 \times 6 = 1679616\)
• \(1679616 \times 6 = 10077696\)

This step-by-step approach might seem tedious but helps in understanding the accumulative nature of exponential growth.

Calculator Use

Using a calculator is an excellent way to verify the results of your repeated multiplication. It's especially useful for checking large computations like \(6^9\). While doing long calculations by hand increases understanding, a calculator ensures accuracy and efficiency.

Here's a simple guide on how to use a calculator for exponential problems:

• Start by entering the base number, 6.
• Locate the exponentiation function on the calculator, often represented as \(^\), \(\, x^y \), or similar symbols.
• Input the exponent, which is 9 in this instance.
• Press equals or compute to get the result, 10077696 in the case of \(6^9\).

By using a calculator, you can quickly verify that your hand solution is correct. Calculators help ease complex calculations, saving both time and mental energy. They are not just for double-checking; they're essential tools in any mathematician's toolkit