Problem 26 - Algebra 1: Concepts and Skills

Question

Evaluate the power. \(10^{3}\)

Step-by-Step Solution

Answer

The value of \(10^{3}\) is 1000.

Step 1: Understand the Power

In the given problem, we have the expression \(10^{3}\). Here, 10 is the base and 3 is the power or exponent. This implies that we must multiply the number 10 by itself 3 times.

Step 2: Perform the Calculation

Let's carry out the multiplication: \(10 * 10 * 10 = 1000\)

Step 3: Write down the Answer

Therefore, \(10^{3} = 1000\).

Key Concepts

Exponentiation

When we tackle the idea of exponentiation, we are diving into one of the fundamental operations in mathematics. This operation tells us how many times to multiply a number by itself. This concept is represented by writing a smaller number, known as the exponent, above and to the right of a normal-sized number, which is the base. For example, in the expression 10^{3}, exponentiation is used to indicate that we need to multiply the base, 10, by itself 3 times.

Exponentiation is a powerful tool because it allows us to express and compute with very large numbers efficiently. It is used across various fields such as science, engineering, and finance to model exponential growth or decay, compound interest, and more. By using exponents, we can simplify complex calculations and make them more manageable.

Base and Exponent

In the world of mathematics, the terms 'base' and 'exponent' play a vital role in the operation of exponentiation. The 'base' is the number that is being multiplied by itself, and the 'exponent', sometimes called the 'power', tells us the number of times the base is used as a factor in the multiplication.

Let's take the expression 10^{3} as an example. The number 10 is our base—it's the number we're going to repeatedly multiply. On the other hand, the number 3 is our exponent—it informs us that there will be three instances of the base multiplied together. It's crucial to grasp this relationship because it underpins how we interpret and carry out exponentiation. By understanding the interplay between base and exponent, we can unlock the meaning behind these expressions and perform calculations with confidence.

Multiplying Bases

Now, let's narrow our focus to a specific case of exponentiation—multiplying bases when they are the same. This is a common scenario and understanding it can simplify many problems. If we have an expression where the same base is used with different exponents, we can apply rules to multiply them efficiently.

For example, the expression a^{m} * a^{n} shows the same base 'a' multiplied but with two different exponents, 'm' and 'n'. The rule here is that when multiplying like bases, we keep the base the same and simply add the exponents together to find the new power. This gives us a^{m+n}. This rule comes in handy especially when dealing with complex expressions involving exponentiation. It's a shortcut that can make calculations much quicker, so it’s well worth remembering!