Problem 2 - Fundamentals of Mathematics

Question

Determine the value of each power and root. \(4^{3}\)

Step-by-Step Solution

Answer

The value of \( 4^3 \) is 64.

Step 1: Understand Exponents

The notation \( 4^3 \) means \( 4 \) raised to the power of \( 3 \). This tells us to multiply \( 4 \) by itself a total of three times. In general, \( a^n \) means the number \( a \) multiplied by itself \( n \) times.

Step 2: Set Up the Multiplication

Write \( 4^{3} \) as a multiplication: \( 4 \times 4 \times 4 \). This expression shows that you need to multiply \( 4 \) by itself twice more after the initial \( 4 \).

Step 3: Perform the Multiplication

Calculate the multiplication step-by-step. First, multiply the first two 4s: \( 4 \times 4 = 16 \). Then, multiply the result by the last 4: \( 16 \times 4 = 64 \).

Key Concepts

Powers

Powers, also known as exponents, are a way to express repeated multiplication of the same number. For instance, the expression \(4^3\) is interpreted as "4 raised to the power of 3." This means you multiply 4 by itself three times: \(4 \times 4 \times 4\). The base is 4, and the exponent or power is 3, indicating the number of times 4 is used as a factor.

• The base: The number being multiplied. Here, the base is 4.
• The exponent: Tells how many times to use the base as a factor. Here, it's 3.
• The result: The final product of the expression. For \(4^3\), this is 64.

Understanding how powers work is crucial for simplifying and solving various mathematical problems. It provides a concise way to represent long multiplications.

Roots

Roots are the inverse operations of powers, serving to "undo" the exponentiation process. For example, if you know \(4^3 = 64\), finding the cube root of 64 will bring you back to the base number 4. In mathematical notation, the square root of a number \(a\) is written as \(\sqrt{a}\), and the cube root as \(\sqrt[3]{a}\).

• The square root \(\sqrt{a}\) asks "what number multiplied by itself gives \(a\)?"
• The cube root \(\sqrt[3]{a}\) asks "what number multiplied by itself three times gives \(a\)?"
• Root calculations help in solving equations involving powers by working backwards to find the original base number.

It's important for solving equations and understanding growth patterns in mathematics, especially in areas like algebra and geometry.

Multiplication

In mathematics, multiplication is a fundamental operation that combines equal groups. When working with powers, multiplication plays a critical role. For example, in \(4^3\), you aren't just adding 4 three times. Instead, you are multiplying it: first \(4 \times 4\) gives 16, and then multiplying that result by another 4 to give 64.

• Multiplication of whole numbers involves adding a number to itself a certain number of times.
• Using multiplication in powers simplifies the calculation process of large numbers.
• It provides a way to understand and calculate large figures efficiently and accurately.

Mastering multiplication is essential as it forms the base for understanding more complex mathematical operations such as exponents and even algebraic expressions.