Problem 62 - Prealgebra

Question

Multiply the decimal by the given power of 10 . \(60.890 \cdot 10^{3}\)

Step-by-Step Solution

Answer

The product is 60,890.

Step 1: Understanding the Problem

We need to multiply the decimal number 60.890 by the power of 10, which is expressed as \(10^3\). This operation will shift the decimal point to the right by three places.

Step 2: Shifting the Decimal

To multiply by \(10^3\), we shift the decimal point in the number 60.890 three places to the right. The number of places is determined by the exponent 3 on the 10.

Step 3: Calculate New Decimal Position

Starting with the original position of the decimal in 60.890 (after 60), move it three places to the right: \(60.890\) becomes \(60890.0\) after moving the decimal.

Key Concepts

Powers of Ten

In mathematics, powers of ten are a handy way to simplify how we write and work with big numbers or small decimals. Instead of writing 1000, we can write it as \(10^3\). This notation tells us that we are multiplying 10 by itself three times, which is \(10 \times 10 \times 10 = 1000\). Powers of ten are particularly useful when dealing with scientific notation or when simplifying arithmetic operations such as multiplication and division with decimals.

A power of ten is expressed as \(10^n\), where **n** is the exponent. The exponent indicates how many times to multiply 10 by itself. Here are some examples:

• \(10^1 = 10\)
• \(10^2 = 100\)
• \(10^3 = 1000\)

By understanding powers of ten, we can easily manipulate decimals and large numbers without manual counting.

Decimal Point Shift

When multiplying a decimal by a power of ten, the decimal point shifts to the right. This rule states that the number of places you move the decimal point is equal to the exponent of 10. For example, multiplying by \(10^3\) means shifting the decimal three places to the right.

Consider the number 60.890.

• When shifted three places to the right due to \(10^3\), it becomes 60890.0.
• The decimal point moves beyond the digits, essentially transforming the decimal into a whole number.
• This ‘shift’ method uses the principles of place value in decimals, where each digit represents a power of ten.

Understanding how the decimal point shifts ensures accurate multiplication with powers of ten without actual calculation.

Exponents in Mathematics

Exponents are a powerful mathematical tool. They allow us to express repeated multiplication succinctly. In any expression such as \(a^b\), 'a' is the base and 'b' is the exponent. Exponents indicate the times a base number is multiplied by itself.

For instance:

• \(3^2\) means \(3 \times 3 = 9\).
• \(5^3\) means \(5 \times 5 \times 5 = 125\).

In the context of multiplying by powers of ten, exponents dictate how far we move the decimal point. The number \(10^3\) instructs us to shift the decimal three places to the right.

Understanding exponents is crucial for tackling decimals and larger numbers across various areas of mathematics, from simple arithmetic to complex algebraic expressions. This fundamental concept underlies a vast array of operations in math, making it a key skill in a student's mathematical toolkit.