Problem 50 - Basic College Mathematics

Question

Write the answer using fraction notation. $$ \left(\frac{1}{2}\right)^{5}\left(\frac{3}{5}\right) $$

Step-by-Step Solution

Answer

\( \frac{3}{160} \)

Step 1: Write the Expression

Start by writing the given expression: \[ \left( \frac{1}{2} \right)^{5} \left( \frac{3}{5} \right) \]

Step 2: Simplify Exponentiation

Simplify the exponentiation part of the expression: \[ \left( \frac{1}{2} \right)^{5} = \frac{1^{5}}{2^{5}} = \frac{1}{32} \]

Step 3: Multiply the Fractions

Now multiply two fractions: \[ \frac{1}{32} \times \frac{3}{5} = \frac{1 \times 3}{32 \times 5} = \frac{3}{160} \]

Key Concepts

fraction notation

Fraction notation is a way of expressing a number that represents a part of a whole or a ratio between two numbers. A fraction consists of two parts: the numerator and the denominator. The numerator is the top number, while the denominator is the bottom number. For example, in the fraction \(\frac{1}{2}\), 1 is the numerator and 2 is the denominator. When writing fractions, it's important to maintain this format to clearly convey the relationship between the two numbers. Proper fraction notation is crucial for correctly solving math problems that involve fractions.

exponentiation

Exponentiation is a mathematical operation that involves raising a number, called the base, to a power, which is the exponent. The exponent indicates how many times the base is multiplied by itself. In the given exercise, we have the fraction \(\frac{1}{2}\) raised to the power of 5. We write this as \(\bigg( \frac{1}{2} \bigg)^5\). To simplify this, we separate the multiplication of the numerator and the denominator: \(\big(1^5 \big) \bigg/ \big(2^5 \big)= \frac{1}{32}\). This tells us that the fraction \(\frac{1}{2}\) is multiplied by itself five times, resulting in \(\frac{1}{32}\).

fraction simplification

Simplifying fractions means reducing them to their simplest form. This involves ensuring the greatest common divisor (GCD) of the numerator and denominator is 1. Let's take the result from our exponentiation \( \frac{1}{32} \times \frac{3}{5} \). We multiply the fractions by multiplying the numerators and the denominators: \(\frac{1}{32} \times \frac{3}{5} = \frac{1 \times 3}{32 \times 5} = \frac{3}{160}\). Here, \( \frac{3}{160} \) is already in its simplest form because the GCD of 3 and 160 is 1. Before considering a multiplication or further operations, always verify if the fraction can be simplified.

multiplying fractions

Multiplying fractions involves multiplying the numerators together and the denominators together. In our given solution, after simplifying the exponentiation, we had to multiply two fractions: \( \frac{1}{32} \times \frac{3}{5} \). To do this, perform the multiplication as follows:
1. Multiply the numerators: \( 1 \times 3 = 3 \).
2. Multiply the denominators: \( 32 \times 5 = 160 \).
This results in the fraction \( \frac{3}{160} \). It's essential to keep the multiplication straightforward by treating it as the product of the numerators over the product of the denominators. Moreover, always check if the resulting fraction can be simplified further.