Problem 74 - Algebra 1

Question

Evaluate the expression. $$\left(\frac{1}{5}\right)^{4}$$

Step-by-Step Solution

Answer

The evaluated expression of \( \left(\frac{1}{5}\right)^{4} \) is \( \frac{1}{625} \)

Step 1: Identify the base and the exponent

In the given expression \(\left(\frac{1}{5}\right)^{4}\), the base is \(\frac{1}{5}\) and the exponent 4. Therefore we must multiply the base 4 times.

Step 2: Multiply the base

As described above, we need to multiply the base, which is \(\frac{1}{5}\) four times: \(\left(\frac{1}{5}\right)^{4} = \frac{1}{5} * \frac{1}{5} * \frac{1}{5} * \frac{1}{5}\).

Step 3: Evaluate the multiplication

On multiplying the fractions we get: \(\frac{1}{5} * \frac{1}{5} * \frac{1}{5} * \frac{1}{5} = \frac{1}{625}\)

Key Concepts

Fractions

Fractions are a way to express a number that is not whole or an integer. A fraction is made up of two numbers, the numerator, and the denominator, separated by a slash. The numerator is the number above the line and represents how many parts we have. The denominator, below the line, shows into how many equal parts the whole is divided. For example, in the fraction \(\frac{1}{5}\), 1 is the numerator indicating one part, and 5 is the denominator, indicating that the whole is divided into five equal parts.

• Fractions can represent portions of a whole, like pieces of a pie or segments of a number line.
• They are helpful in dividing things into parts and comparing different quantities.
• Fractions can also be used in operations such as addition, subtraction, multiplication, and division.

When multiplying fractions as in our original exercise, the operation signifies how many times you take a fraction of a fraction, essentially reducing the size with each multiplication.

Multiplication

Multiplication is one of the basic arithmetic operations that combines groups of equal sizes. In terms of fractions, multiplication is a bit special as it involves multiplying both numerators and denominators. When multiplying fractions:

• You multiply the numerators together to get the new numerator.
• You multiply the denominators to get the new denominator.

For example, in \(\frac{1}{5} \times \frac{1}{5} \times \frac{1}{5} \times \frac{1}{5}\), the numerators \(1 \times 1 \times 1 \times 1\) give 1, while the denominators \(5 \times 5 \times 5 \times 5\) give 625.
The result is \(\frac{1}{625}\). Multiplication makes sense as combining fractions iteratively gives fractions of fractions, making the result smaller very quickly, especially with the repeated multiplication of a number less than one. Understanding multiplication helps in grasping how to work with repeated multiplication (as done in powers) effectively.

Powers

When dealing with exponents, such as in \(\left(\frac{1}{5}\right)^4\), you are addressing the concept of powers in mathematics. A power consists of a base and an exponent.

• The base is the number or expression being multiplied.
• The exponent shows how many times to multiply the base by itself.

In our exercise, \(\left(\frac{1}{5}\right)^4\) indicates the fraction \(\frac{1}{5}\) is to be multiplied by itself four times. This is why we wrote it as \(\frac{1}{5} \times \frac{1}{5} \times \frac{1}{5} \times \frac{1}{5}\).
Each time you multiply another \(\frac{1}{5}\), you are taking a fraction of the previous fraction, which makes the number rapidly decrease in size. Understanding powers is crucial for simplifying expressions and is used extensively in algebra and calculus. It helps visualize multiplication's effect growing or shrinking numbers exponentially based on repeated operations.