Problem 52 - Fundamentals of Mathematics

Question

Determine the value of each of the powers. Use a calculator to check each result. \(5^{8}\)

Step-by-Step Solution

Answer

The value of \(5^8\) is 390625.

Step 1: Understand the Base and Exponent

In the expression \(5^8\), the number 5 is called the base, and the number 8 is the exponent. This means that we need to multiply the base, which is 5, by itself a total of 8 times.

Step 2: Multiply the Base Repeatedly

To calculate \(5^8\), we perform the multiplication: \(5 \times 5 \times 5 \times 5 \times 5 \times 5 \times 5 \times 5\).

Step 3: Evaluate the Power Using a Calculator

Using a calculator to perform the repeated multiplication quickly, we find \(5^8 = 390625\). Enter 5 and press the exponentiation key (usually labeled as '^' or 'EXP'), then enter 8 to find the result.

Key Concepts

Base and Exponent

In mathematics, understanding base and exponent is key when it comes to exponentiation. Let's take a closer look at the components of the expression \(5^8\). In this expression, the base is 5, and the exponent is 8. The base is the number that you will multiply by itself. The exponent tells you how many times to use the base as a factor in your multiplication. So when you see an expression like \(a^b\), it essentially means that the base \(a\) is multiplied by itself \(b\) times. This is a concise way to express repeated multiplication, making calculations much more efficient.
By defining the base and exponent in this way, we simplify complex multiplication into easily manageable expressions. This becomes particularly helpful when dealing with larger numbers, as it allows us to use a compact form instead of writing out each multiplication operation.

Power

A power in mathematics refers to the result of exponentiation. When we talk about the power of a number, we are referring to the result you get after raising a base to an exponent. For example, in our original problem \(5^8\), the power is the result of multiplying the base, 5, by itself 8 times. This calculated power is 390,625. In general, powers make it easier to represent and work with very large or very small numbers and are fundamental in various fields, such as science and engineering.
Using powers, we often express numbers in scientific notation, which helps in managing very big or very small quantities concisely. Powers are key to understanding exponents, as learning how they function can enhance your comprehension of mathematics profoundly.

• Powers help simplify expressions.
• They are the result of exponent operations.
• Essential in scientific and engineering calculations.

Multiplication

Multiplication is a fundamental arithmetic operation that's sometimes referred to as repeated addition. When you exponentiate, like in \(5^8\), you perform a form of repeated multiplication. Here, you take the number 5 and multiply it by itself 8 times. Understanding multiplication is integral because it serves as the building block for understanding how exponentiation works.
For example, multiplying 5 by itself four times is represented as \(5 \times 5 \times 5 \times 5\), but when dealing with enormous values or frequent calculations, using exponents is far more efficient. Multiplication, especially in the context of exponents, allows us to break down complex problems into simpler tasks.

• Foundation for learning exponents.
• Makes operations quicker and easier through repeated operations.
• Vital for expressing longer calculations succinctly.