Problem 59 - Prealgebra

Question

Multiply the decimal by the given power of 10 . \(53.867 \cdot 10^{4}\)

Step-by-Step Solution

Answer

The result is 538670.

Step 1: Understand the Problem

We need to multiply the decimal number, 53.867, by a power of 10, specifically by \(10^4\). This involves moving the decimal point to the right by 4 places.

Step 2: Recognize the Power of 10

The expression \(10^4\) means the number 10 raised to the fourth power or 10000. In decimal multiplication, multiplying by \(10^n\) moves the decimal point \(n\) places to the right.

Step 3: Move the Decimal Point

Move the decimal point in 53.867 four places to the right to perform the multiplication. You will add zeros as needed to maintain the value of the number.

Step 4: Shift the Decimal

Starting at 53.867: 1. Move the decimal one place right: 538.67 2. Second move: 5386.7 3. Third move: 53867. 4. Fourth move: 538670. The new number is 538670.

Key Concepts

Decimal point

Decimals are numbers that include fractional parts, separated from the whole number part by a decimal point. The decimal point is essential in understanding where the fractional part begins. It acts as a marker to differentiate between whole numbers and parts of a number less than one. For instance, in the number 53.867, the digits 53 are the whole number part, and 867 is the fractional part.

When performing operations such as multiplication by powers of 10, understanding the role of the decimal point is crucial. The position of this point determines the magnitude of the number. In essence, moving the decimal point to the right increases the number by a factor of 10 for each place moved.

Thus, in the operation involving decimal multiplication by powers of 10, the decimal point acts as the guide to ensure that you maintain the correct value of the number.

Powers of 10

Powers of 10 are a way of expressing large numbers in a simplified form using exponents. An exponent indicates how many times a number is multiplied by itself. For example, the expression \(10^4\) means multiplying 10 by itself four times, equalling 10,000. This is a potent concept in math, especially when dealing with decimals.

When you multiply a decimal by a power of 10, you essentially shift the decimal point to the right by the number of times equal to the power. For instance, multiplying 53.867 by \(10^4\) means moving the decimal point four places to the right.

This operation helps to scale up the number quickly and is particularly useful in ensuring precision in calculations involving decimals.

Place value

Place value refers to the value assigned to each digit based on its position in a number. This concept is fundamental when working with decimals as it helps to understand the significance of each numeral within a number.

In the decimal number 53.867:

• 5 is in the tens place, contributing 50 to the number.
• 3 is in the ones place, contributing 3 to the number.
• 8 is tenths, 6 is hundredths, and 7 is thousandths, contributing fractional parts to the number.

When you multiply by a power of 10, each digit's place value is multiplied by 10, effectively moving it to a higher position. Moving the decimal point to the right enhances the place value of each digit, turning fractional parts into whole number parts as necessary.

Recognizing place value helps in not losing the mathematical integrity of a number during such operations.