Problem 47 - Prealgebra

Question

Simplify each of the following as much as possible, and write all answers as decimals. $$\left(\frac{1}{3}\right)^{2}(5.4)+\left(\frac{1}{2}\right)^{3}(3.2)$$

Step-by-Step Solution

Answer

The simplified decimal value of the expression is 1.0.

Step 1: Simplify each fraction within their powers

Calculate \( \left( \frac{1}{3} \right)^2 \) and \( \left( \frac{1}{2} \right)^3 \). First, we find \( \left( \frac{1}{3} \right)^2 = \frac{1}{3} \times \frac{1}{3} = \frac{1}{9} \). Next, we calculate \( \left( \frac{1}{2} \right)^3 = \frac{1}{2} \times \frac{1}{2} \times \frac{1}{2} = \frac{1}{8} \).

Step 2: Multiply each resulting fraction with respective numbers

Multiply \( \frac{1}{9} \) by 5.4 and \( \frac{1}{8} \) by 3.2. Perform the multiplication for each: \( \frac{1}{9} \times 5.4 = 0.6 \) and \( \frac{1}{8} \times 3.2 = 0.4 \).

Step 3: Add the results together

Add the results from the two multiplications: \( 0.6 + 0.4 = 1.0 \). This is the final result of the expression simplified and written as a decimal.

Key Concepts

Decimal Conversion

Understanding decimal conversion is essential for simplifying expressions and is a key skill in mathematics. A decimal is simply a way to express fractions or whole numbers in a base-10 system. In problems involving decimal conversion, you'll typically change fractions into their decimal equivalents for easier calculations. For instance, when you need to multiply a fraction by a decimal or add decimals together, converting to decimals often simplifies the process.

• To convert a fraction to a decimal, you divide the numerator by the denominator. For example, to convert \(\frac{1}{3}\), divide 1 by 3 to get approximately 0.333.
• Understanding decimal conversion will help you perform operations like addition, subtraction, multiplication, and division more efficiently, as it aligns with the base-10 number system we use daily.

Knowing how to convert between fractions and decimals expands your mathematical toolbox, allowing you to solve a wide range of problems with ease.

Fractional Exponents

Fractional exponents represent the power to which a number is raised, with both integers and fractions as possible exponents. They are a handy way to express roots using powers. For example, the fraction \(\frac{1}{3}\), when used as an exponent, expresses that you are cubing a number. In the context of our exercise, raising a fraction to a power is a means of simplifying it.

• For \(\left(\frac{1}{3}\right)^2\), the base fraction \(\frac{1}{3}\) is simply multiplied by itself: \(\frac{1}{3} \times \frac{1}{3} = \frac{1}{9}\).
• Similarly, \(\left(\frac{1}{2}\right)^3\) involves multiplying \(\frac{1}{2}\) three times: \(\frac{1}{2} \times \frac{1}{2} \times \frac{1}{2} = \frac{1}{8}\).

This makes solving expressions involving fractional exponents clearer, simplifying your work by consistently applying the multiplication rule.

Multiplication of Fractions

Multiplication of fractions is a foundational concept that involves multiplying the numerators and denominators of the given fractions to find the product. This method is straightforward, as it simplifies expressions effectively.

• To multiply a fraction by a number, or by another fraction, multiply the numerators together, do the same with the denominators to keep it streamlined.For instance, multiplying \(\frac{1}{9}\) by 5.4 involves initially understanding it as: \(\frac{1}{9} \times \frac{27}{5} = \frac{27}{45} = 0.6\).
• The critical step involves simplification, where the fraction result is converted into a decimal to get easier reading and understanding.

Whether you’re working with just fractions or incorporating decimals, mastering their multiplication keeps math simpler and quicker.

Addition of Decimals

Adding decimals is a common operation that we've practiced often, probably without realizing it. To add decimals correctly, ensure that the decimal points are aligned in accordance with each other, which guarantees accuracy in your results.

• The simpler example from our exercise demonstrates this principle well: adding 0.6 and 0.4. When they are placed one under the other, with decimal points lined up, their addition becomes direct and exact.
• Decimal addition involves calculating the sum of each column, carrying over when the sum exceeds 9, similar to whole number addition in principle. This fundamental concept helps in dealing with mixed numbers and other complex calculations.

Mastering addition of decimals allows effectively tackling a range of mathematical problems, making you more confident when handling larger, more complex sets of numbers.