Problem 75 - Prealgebra

Question

Simplify. $$4^{3}$$

Step-by-Step Solution

Answer

\(4^3 = 64\).

Step 1: Understand Exponents

Exponents are used to express repeated multiplication of the same number. The expression \(4^3\) means the number 4 is multiplied by itself 3 times.

Step 2: Expand the Exponential Expression

Write the exponential expression as a repeated multiplication: \(4^3 = 4 \times 4 \times 4\).

Step 3: Multiply the First Two Factors

First, multiply the first two factors: \(4 \times 4 = 16\).

Step 4: Multiply the Result with the Last Factor

Take the result from the previous step and multiply it with the last factor: \(16 \times 4 = 64\).

Step 5: Simplified Result

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

Key Concepts

Simplifying Exponents

When you encounter an exponent, it means you're dealing with a shorthand for repeated multiplication. Simplifying exponents helps to express this repeated multiplication in a more manageable form. For example, instead of writing out the multiplication each time, you simply use the exponent to know how many times to multiply a number by itself.

Let's look at the example of simplifying the exponential term \(4^3\). The base, 4, is your number to multiply, and the exponent, 3, tells you how many times to multiply it by itself. Thus, \(4^3\) indicates you multiply 4 by itself three times. Simplifying exponents is about carrying out these multiplications to arrive at a single numerical result. For \(4^3 = 4 \times 4 \times 4\), the expression simplifies to 64. This helps streamline calculations and makes it simpler to work with bigger numbers without writing out every multiplication instance.

Repeated Multiplication

Exponents are closely tied to the idea of repeated multiplication. They offer a neat and efficient way to show the multiplication of a number by itself multiple times.

In our exercise example, \(4^3\) translates to multiplying 4 by itself three times: \(4 \times 4 \times 4\). Thinking about it step-by-step:

• Start by performing the multiplication of the first two numbers: \(4 \times 4 = 16\).
• Then take that result and multiply by the base number once more: \(16 \times 4 = 64\).

This breakdown into simpler steps highlights the essence of exponents, reinforcing that they inherently represent repeated multiplication. It's a powerful tool for simplifying expressions that may otherwise seem complicated.

Power of a Number

The 'power of a number' term refers to the result you get when you multiply a number by itself numerous times as specified by the exponent. In our specific context, it's about understanding not just the process, but the end result of what's commonly called "raising a number to a power."

In \(4^3\), the base number, 4, is raised to the power of 3. Upon performing the necessary multiplications: \(4 \times 4 \times 4\), you get the power of the number, which is 64. This simplification tells us that 4 raised to the third power (or "cubed") equals 64.

Powering numbers play a crucial role in mathematics, especially when dealing with large computations. The power of a number helps in representing these calculations succinctly, giving clarity to expressions involving multiple layers of multiplication.