Problem 70 - Fundamentals of Mathematics

Question

Find the roots (using your knowledge of multiplication). Use a calculator to check each result. \(\sqrt[4]{256}\)

Step-by-Step Solution

Answer

The fourth root of 256 is 4.

Step 1: Understanding the Problem

We are asked to find the fourth root of 256, which means we need to determine the number that, when multiplied by itself four times, equals 256.

Step 2: Identifying Known Values

We know from simple multiplication tables and powers of numbers that 4 to the power of 4 (i.e., 4 * 4 * 4 * 4) is a common power calculation. Let's check this value.

Step 3: Performing Multiplication

Calculate 4 multiplied by itself four times:\[ 4 \times 4 = 16 \]\[ 16 \times 4 = 64 \]\[ 64 \times 4 = 256 \]This confirms that 4 to the power of 4 is 256.

Step 4: Verifying with a Calculator

To ensure the calculation is correct, use a calculator to input the number 4 and raise it to the 4th power (4^4) to confirm it equals 256.

Key Concepts

Fourth Roots

The concept of a fourth root is integral when dealing with powers and exponents. A fourth root of a number is a value that, when multiplied by itself four times, gives the original number. In mathematical terms, if we want to find the fourth root of 256, we are searching for a number such that \(n^4 = 256\). The fourth root is like the inverse operation of raising a number to the power of four. This means if you know one operation, finding the other becomes easier.

• Fourth roots are symbolic and often written as \(\sqrt[4]{x}\).
• They provide a way to simplify the understanding of complex numbers.

For the integer 256, recognizing that it's a power of another base number (in this case, 4) is essential. Once you identify this, you can perform simple verifications using \(4^4 = 256\) to confirm your understanding.

Multiplication

Multiplication is a core mathematical operation that students meet early on, but its applications show up in advanced topics like finding roots or exponents. When you learn about fourth roots, the process heavily involves multiplication, as it demands checking the product of a number multiplied by itself multiple times. Here, multiplication steps help verify the result.

• Multiplication of the same number is often seen in calculating higher powers.
• Keeps the underlying arithmetic straightforward.
• Supports understanding of more complex algebraic operations.

In our specific example involving \(\sqrt[4]{256}\), multiplication is performed as follows: \(4 imes 4 = 16\), then \(16 imes 4 = 64\), and finally \(64 imes 4 = 256\). This sequence verifies that the fourth root of 256 is indeed 4.

Powers of Numbers

Understanding powers of numbers is crucial in mathematics, especially when finding roots, such as fourth roots. A power refers to how many times a number (the base) is multiplied by itself. When we say \(4^4\), it suggests that the number 4 is used as a factor four times.

• Powers like \(n^4\) indicate repeated multiplication: n times n times n times n.
• They enable simplification of large numbers and expressions.

For example, knowing \(4^4 = 256\), helps in deducing that the fourth root of 256 is 4—since reversing the process of moving from base to power leads back to the fourth root. Powers allow us to express and work with very large numbers succinctly, making calculations less cumbersome.