Problem 17 - Elementary and Intermediate Algebra

Question

Simplify. $$ 4^{3} $$

Step-by-Step Solution

Answer

64

Step 1: Understand the Base and the Exponent

The base is 4 and the exponent is 3. This means multiplying the base, 4, by itself 3 times.

Step 2: Multiply the Base by Itself

First multiply 4 by 4: equation\[4 \times 4 = 16\]

Step 3: Multiply the Result by the Base Again

Next, take the result (16) and multiply it by 4 again: equation\[16 \times 4 = 64\]

Step 4: Write the Final Answer

The simplified form of \(4^3\) is 64.

Key Concepts

Base and Exponent

In exponentiation, we work with two main components: the base and the exponent. The base is the number that will be multiplied, and the exponent tells us how many times to multiply the base by itself.
In our example with the expression \(4^3\), the base is 4, and the exponent is 3. This means that we will multiply 4 by itself three times. Understanding these two elements is crucial for working with exponents in any type of math problem.

Multiplication

Multiplication in exponentiation involves repeatedly multiplying the base by itself as many times as the exponent indicates. Let’s break down the multiplication steps.
First, we multiply the base by itself once: \[4 \times 4 = 16\]
This gives us the intermediate result of 16.
Next, we take this result and multiply it by the base again: \[16 \times 4 = 64\]
So, by multiplying step-by-step, we can simplify \(4^3\) to 64. Breaking down each step like this helps in understanding how multiplication works within exponentiation.

Simplifying Exponents

Simplifying exponents means reducing an expression involving exponents to its simplest form. In the example \(4^3\), we work through the multiplication steps until we reach a single number.
Initially, we have \(4^3\), which translates to multiplying 4 by itself three times: \[4 \times 4 \times 4\]
We then simplify step-by-step. First, multiply the base by itself once to get an intermediate value: \[4 \times 4 = 16\]
Finally, multiply this result (16) by the base one more time: \[16 \times 4 = 64\]
Thus, \(4^3\) simplifies to 64. This process shows how breaking down and understanding each step simplifies the expression efficiently.