Question

Use a calculator to evaluate $$4^{3}$$

Short answer

The calculated value of \(4^{3}\) is 64

Step-by-step solution

Understanding Exponentiation

Exponentiation is an operation on two numbers. In this case, 4 and 3. The number 4 is the base and 3 is the exponent. When a number is raised to the power of 3, this means that you have to multiply the number by itself twice.

Calculate

To evaluate \(4^{3}\), multiply 4 by itself twice. That is \(4 * 4 * 4\).

Final Calculation

After carrying out the multiplication, \(4 * 4 * 4\) equals 64.

Key concepts

Power of a Number

When learning about the power of a number, you’re looking at what happens when a number, known as the base, is raised to an exponent. This is a form of shorthand to express repeated multiplication of the same factor. For instance, if we take the number 5, and we want to raise it to the power of 3, written as \( 5^3 \), what we’re actually doing is multiplying the number 5 by itself three times, like this: \( 5 \times 5 \times 5 \).

Understanding the power of a number can be intuitive by visualizing groups of the base number. If you had 5 groups of 5 blocks, you'd quickly realize that this is the same as \( 5^3 \), or 125 blocks. This visualization helps clarify that the exponent just tells you how many times to use the base as a multiplier.

Evaluate Exponents

To correctly evaluate exponents, it's important to understand the order of operations and to carefully follow it. For a number raised to an exponent, like \( 6^2 \), you need to recognize that this does not mean 6 times 2, but rather 6 times itself, or \( 6 \times 6 \), which equals 36.

When evaluating exponents, it's also important to remember that any number (except zero) raised to the power of zero is 1. This is because the exponent indicates the number of times you use the base as a multiplier, and if you're not using the base at all (zero times), you're essentially left with a single instance of the number 1. For example, \( 2^0 = 1 \).

It's common for students to rush through these calculations, but taking them step by step will ensure accuracy and a real understanding of how the exponents affect the base number.

Exponential Expressions

An exponential expression is a mathematical phrase that involves exponents. The basic form of an exponential expression is a base raised to an exponent. The process of multiplying the base by itself the number of times indicated by the exponent is what defines exponential growth, a rapid increase in value that can be visualized on a graph as a steep curve.

When working with exponential expressions, especially in advanced mathematics, you might encounter different bases and exponents, including fractions and negative numbers. It's pivotal to remember that negative exponents translate to the reciprocal of the base raised to the positive exponent. For example, \( 3^{-2} = \frac{1}{3^2} = \frac{1}{9} \).

Tips for Success

• Always write out the full expression, showing each repetition of the base
• For larger exponents, you may use a calculator but understand the concept first
• Practice with different bases, including fractions and negatives, to build a strong foundation

Through practice and understanding, exponential expressions become a powerful tool in algebra and beyond, enabling you to succinctly write and calculate with very large or very small numbers.