Problem 44 - Prealgebra

Question

The problems below are a review of some of the concepts we covered previously. Find the following products. (Multiply.) $$(0.7)^{2}$$

Step-by-Step Solution

Answer

The product of \((0.7)^2\) is 0.49.

Step 1: Identify the Problem

We need to find the square of 0.7, which means multiplying 0.7 by itself.

Step 2: Write the Mathematical Expression

Represent the problem using mathematical notation: \((0.7)^2\).

Step 3: Perform the Multiplication

Calculate the product of 0.7 multiplied by 0.7. Use standard multiplication rules, or a calculator, to find \(0.7 \times 0.7 = 0.49\).

Step 4: Verify with Squaring Rules

Since squaring any number involves multiplying it by itself, ensure 0.49 makes sense by recalling that the square of a number less than 1 will be smaller than the original number.

Key Concepts

Understanding Multiplication

Multiplication is one of the basic arithmetic operations. In simple terms, it's a way of adding a number to itself repeatedly. For instance, if you see \(3 \times 4\), it means you add the number 3, four times:

• 3 + 3 + 3 + 3 = 12

The number of times you add the number (4 in this case) is called the "multiplier," and the number you add (3 here) is the "multiplicand." Then, 12 is the "product." This same operation applies to all numbers, whether whole, fractions, or decimals.

If multiplying decimals like 0.7 \( \times \) 0.7, it means to calculate 0.7 added to itself 0.7 times in a conceptual sense. Don't worry; you just treat it like any other multiplication at its core, focusing on the place values of the numbers involved.

Delving into Decimals

Decimals are numbers that are divided into parts or fractions of 10, 100, 1000, and so on. They have a dot called the decimal point. The position of a digit in a decimal number determines its value. For example in 0.7 the "7" is in the tenths place, representing 7 parts of 10, or \(\frac{7}{10}\).

When multiplying decimals, you need to pay close attention to the decimal places. For example, while multiplying 0.7 \( \times \) 0.7, first multiply the numbers as if they were whole numbers: 7 \( \times \) 7 = 49.

Then, count the decimal places: there are two, one from each 0.7. So, you place the decimal point in the product, 49, moving it two places from the right to get 0.49.

• Understanding place value is key.
• Place your decimal correctly in your answers.

Exploring Exponents

Exponents are a way of expressing repeated multiplication of the same number. For example, \(2^3\) means you multiply 2 by itself three times:

• 2 \( \times \) 2 \( \times \) 2 = 8

Here, the number 2 is the "base," and 3 is the "exponent." When the exponent is 2, like in \((0.7)^2\), it's known as "squaring." This term comes from the idea of a square where each side is of equal length.

When squaring decimals, be cautious about place-value changes, just as in regular multiplication. A useful thing to remember is:

• Squaring numbers less than 1 gives results less than the given number.
• Formula: \( x^2 = x \times x \).