Problem 22 - Fundamentals of Mathematics

Question

Find the value of each of the following. $$\left(\frac{3}{10}\right)^{2}$$

Step-by-Step Solution

Answer

\( \left(\frac{3}{10}\right)^{2} = \frac{9}{100} \).

Step 1: Identify the base and exponent

In the expression \( \left(\frac{3}{10}\right)^{2} \), the base is a fraction, \( \frac{3}{10} \), and the exponent is 2. This means we have to multiply the base by itself.

Step 2: Multiply the base by itself

Since the exponent is 2, calculate \( \frac{3}{10} \times \frac{3}{10} \). When multiplying fractions, multiply the numerators and the denominators separately.

Step 3: Multiply the numerators

Calculate the numerator of the product by multiplying \(3 \times 3 = 9\).

Step 4: Multiply the denominators

Calculate the denominator of the product by multiplying \(10 \times 10 = 100\).

Step 5: Write the final answer

Combine the results from Step 3 and Step 4. The product is \( \frac{9}{100} \). Thus, \( \left(\frac{3}{10}\right)^{2} = \frac{9}{100} \).

Key Concepts

Multiplying Fractions

When dealing with fraction exponents, it's important to understand how to multiply fractions correctly. Multiplying fractions involves taking two fractions and finding their product. This is done by multiplying the numerators together and the denominators together. This way, you'll have a new fraction that represents the product of the two.

Let's consider an example: if you have the expression \( \left( \frac{3}{10} \right)^{2} \). This means you will need to multiply \( \frac{3}{10} \) by itself. To do this:

• Multiply the numerators: 3 \( \times \) 3 gives 9
• Multiply the denominators: 10 \( \times \) 10 gives 100

The result is \( \frac{9}{100} \), showcasing how multiplying fractions operates by addressing both the top and bottom numbers separately.

Base and Exponent

In mathematical expressions, especially those involving fractions, it's essential to clearly identify the base and the exponent. The base is the underlying number or expression we are raising to a power, and the exponent indicates how many times to multiply the base by itself.

For instance, if we look at \( \left(\frac{3}{10}\right)^{2} \), the fraction \( \frac{3}{10} \) serves as the base. The exponent, in this case, is 2, which tells you to multiply \( \frac{3}{10} \) by itself once more. Thus, it invites you to think of the process as repetitive multiplication guided by the exponent value. Understanding this concept is crucial for solving any exponent problem effectively.

Numerator and Denominator Multiplication

When multiplying fractions, it's critical to treat the numerators and denominators separately. The numerator is the top number of the fraction, which represents how many parts we have. The denominator, on the other hand, is the bottom number, which states how many parts make up the whole.

In our exercise, the multiplication \( \frac{3}{10} \times \frac{3}{10} \) clearly demonstrates this separation:

• The numerators 3 and 3 are multiplied to get 9.
• The denominators 10 and 10 are multiplied to get 100.

Thus, the final expression becomes \( \frac{9}{100} \). By consistently applying these simple multiplication rules, you gain the ability to handle any fraction multiplication challenge with confidence. Recognizing this process ensures a solid grasp on one of the fundamental operations within mathematics.