Problem 22 - Algebra 1

Question

DECIMAL FORM Rewrite in decimal form. $$ 4.332 \times 10^{8} $$

Step-by-Step Solution

Answer

The decimal form of \( 4.332 \times 10^{8} \) is 433200000.0.

Step 1: Identify the base and exponent

In the given scientific notation, 4.332 is the base and 8 is the exponent. This means \( 4.332 \times 10^{8} \).

Step 2: Expand the scientific notation using exponent rule

According to exponent rule, multiplying a number by \( 10^{n} \) simply moves the decimal point of the number n places to the right. Therefore, here, the 8 in \( 10^{8} \) is indicating that the decimal of the base number (4.332) needs to be moved 8 places to the right.

Step 3: Convert the scientific notation to a decimal number

By moving the decimal point to the right 8 times, we get the decimal form of the given scientific notation. But along the way, we must add zeros in any empty spaces that we have created by moving the decimal.

Key Concepts

Decimal Numbers

Decimal numbers are a way to express fractions and whole numbers together using a decimal point. In mathematics, a decimal number is a numerical notation that includes a whole number and a fractional part separated by a decimal point. The digits following the decimal point show a value smaller than one.

For example, in 4.332, 4 is the whole number, and .332 is the fractional part. Decimal numbers make it easy to work with non-whole numbers in calculations, giving you a clear understanding of the size and value of numbers at a glance.

Key characteristics of decimal numbers include:

• The position of each digit determines its value, similar to whole numbers but also extending beyond the decimal point.
• Decimals enable precise representation of values that are not whole, giving a more accurate numerical representation.​

Understanding how decimals work with other mathematical concepts, like scientific notation, is crucial to solving problems efficiently.

Exponents

Exponents are a mathematical notation for expressing repeated multiplication of a number by itself. The exponent tells us how many times the base number is used as a factor in the multiplication. Exponents make calculations more manageable and provide a shorthand method to express large numbers.

In the expression \(10^{8}\), the numeral 10 is the base, and 8 is the exponent. This indicates that 10 should be multiplied by itself 8 times, leading to a large result, 100,000,000.

When working with exponents:

• An exponent of 2 signifies the square of a number \( (x^2 = x \times x) \).
• An exponent of 3 signifies the cube \( (x^3 = x \times x \times x) \).
• A positive exponent denotes repeated multiplication, while a negative exponent suggests repeated division.

This understanding of exponents is exceptionally useful for converting scientific notation to decimal forms and simplifying complex calculations.

Base and Exponent

The terms 'base' and 'exponent' work together to define the power of a number. In the context of an exponential expression like 4.332 \( \times 10^8 \), 4.332 is considered the base, and 8 is the exponent.

• Base: This is the number that is being multiplied. It could be any real number, depending on the problem at hand. In scientific notation, the base represents the significant digits of the number.
• Exponent: This represents how many times the base should be multiplied by itself. Here, the number 8 signifies that 10 will be used as a multiplier eight times.

Together, base and exponent allow us to express very large or very small numbers succinctly through scientific notation, helping to maintain accuracy without writing lengthy sequences of zeros or decimals.