Problem 48 - Prealgebra

Question

Simplify each of the following as much as possible, and write all answers as decimals. $$\left(\frac{1}{5}\right)^{2}(7.5)+\left(\frac{1}{4}\right)^{2}(6.4)$$

Step-by-Step Solution

Answer

The simplified result is 0.7.

Step 1: Simplify the first expression

Calculate \( \left(\frac{1}{5}\right)^{2} \), which is \( \frac{1}{25} \). Then, multiply the result by 7.5: \( \frac{1}{25} \times 7.5 = 0.3 \).

Step 2: Simplify the second expression

Calculate \( \left(\frac{1}{4}\right)^{2} \), which is \( \frac{1}{16} \). Then, multiply the result by 6.4: \( \frac{1}{16} \times 6.4 = 0.4 \).

Step 3: Add the results from Step 1 and 2

Add the two decimal results together: \( 0.3 + 0.4 = 0.7 \).

Key Concepts

Simplifying Expressions

Simplifying expressions is a fundamental skill in mathematics that involves reducing expressions into their simplest form. This can make it easier to understand and work with them. In this context, there are a few basic steps you can follow when simplifying expressions involving fractions and decimals:

• Compute powers or exponents, if any are present.
• Multiply or divide fractions and decimals as needed.
• Add or subtract the results to get the simplest form.

By following these steps, you reduce the complexity of a given mathematical expression. In our example, this process involved calculating the power of fractions and then multiplying them by a decimal. Breaking down the calculations step-by-step helped achieve the simplest decimal form for further calculations.

Fractions

Fractions are a way to represent numbers that are not whole. A fraction consists of two parts: the numerator on top and the denominator on the bottom, separated by a line. A fraction denotes the division of two numbers, which can be expressed as:\[\frac{a}{b}\]Where \(a\) is the numerator and \(b\) is the denominator.Fractions can be simplified or transformed depending on the operation required. In the exercise, we squared fractions like \(\left(\frac{1}{5}\right)^2\), meaning the fraction is multiplied by itself. To do this, both the numerator and the denominator are individually squared, converting them into simpler terms:\[\left(\frac{1}{5}\right)^2 = \frac{1^2}{5^2} = \frac{1}{25}\]Understanding fractions and their transformations are crucial for performing algebraic operations accurately. This knowledge helped us convert fractions directly for further multiplication by decimals.

Exponents

Exponents are a shorthand way of expressing repeated multiplication. When you see a number raised to an exponent, it indicates how many times the number is to be multiplied by itself. For example, \(2^3\) means \(2 \times 2 \times 2\), resulting in 8.Exponents also apply to fractions, requiring each component (numerator and denominator) to be multiplied individually. For instance, with the fraction \(\frac{1}{5}\):\[\left(\frac{1}{5}\right)^2 = \frac{1^2}{5^2} = \frac{1}{25}\]Exponents make it easy to simplify expressions before engaging in other operations like multiplication or addition. When working with exponents, it is important to remember the basic laws:

• Multiplying powers with the same base adds the exponents: \(a^m \times a^n = a^{m+n}\).
• Raising a power to another power multiplies the exponents: \((a^m)^n = a^{m \times n}\).
• Any number to the power of zero is one: \(a^0 = 1\).

Recognizing and applying these laws aids in simplifying and solving expressions efficiently, ensuring our final expression is uncomplicated and easy to interpret.