Question

Compute ([7]73) \(^{15}\) by raising 7 to the \(15^{\text {th }}\) power

Short answer

The result of \(7^{15}\) is quite large, so it's best to use a calculator for the exact value. If done correctly, one will get \(7^{15} = 4747561509943\).

Step-by-step solution

Understanding the power notation

The term \(7^{15}\) represents the number 7 multiplied by itself 15 times.

Performing the exponentiation operation

To compute the value of \(7^{15}\), one needs to multiply 7 by itself 15 times, which can either be done by long multiplication or using a calculator to ensure accuracy and efficiency.

Key concepts

Power notation

Did you know that power notation is simply a way of expressing repeated multiplication of the same number? Here's how it works. When you see something like \(7^{15}\), it means you are taking the number 7 and multiplying it by itself a total of 15 times. In simple terms, think of it as stacking 7s all together, repeatedly multiplying them step by step. Power notation helps us write these large and repeated multiplications more conveniently. Instead of writing 7 x 7 x 7... fifteen times, we can write \(7^{15}\), and it clearly shows the base, which is 7, and the exponent, which is 15.

The base is the number that's being repeatedly multiplied, while the exponent tells us how many times to multiply the base by itself. If we encounter \(a^b\), the formula is to multiply the base \(a\) by itself \(b\) times. This shorthand is particularly useful when dealing with large numbers or high exponents.

Multiplication

Multiplication is all about combining equal groups together. Just like addition, but faster when numbers repeat. When we're dealing with exponentiation, multiplication becomes a big part of the process. Let's dive deeper into what happens with multiplication when we compute powers.

In the exercise \(7^{15}\), we're essentially multiplying 7 over and over. First by itself, then by the product, and continuing until we've done this 15 times overall. This is what we call sequential multiplication. It makes sure every step is built on the last, leading to a very large final product.

Here's a simple way to understand it:

• Start with the base, which is 7.
• First step: 7 x 7 = 49.
• Second step: 49 x 7 = 343.
• Continue this, multiplying the last result by 7, till the 15th time.

By breaking it down like this, you can see how the multiplication slowly builds up, leading to a massive number by the end.

Calculator use

A calculator is a super handy tool when it comes to big computations like exponentiation. Doing \(7^{15}\) by hand would be tedious and can lead to errors. A calculator can do it quickly and accurately. Let's look at how you can utilize it for exponentiation.

Most modern calculators have an exponentiation function. Typically, you'll either see a button labeled as \(^\wedge\) or \("x^{y}"\), which allows you to input the base followed by the exponent. Here's how you can do it:

• Enter the base number, 7.
• Press the exponentiation button (often ^ or x^y).
• Enter the exponent number, 15.
• Hit the equals button to get the result.

By using a calculator, not only do you save time, but you also minimize mistakes that are common in manual calculations. With this method, you can instantly compute not just small exponents, but very large ones too, opening doors to explore more advanced mathematics confidently.