Problem 24 - Algebra 1: Concepts and Skills

Question

Write the number in decimal form. $$ 7.75 \times 10^{0} $$

Step-by-Step Solution

Answer

The decimal form of 7.75 x \(10^{0}\) is 7.75.

Step 1: Identify the parts of the scientific notation

Here 7.75 is the decimal part, and 10^{0} is the scientific notation part. In general, any number raised to the power of 0 is 1.

Step 2: Multiply the decimal part by 10^{0}

Multiply 7.75 by \(10^{0}\) or 1. The result is 7.75.

Key Concepts

Understanding Decimal Form

In mathematics, when we talk about **decimal form**, we're referring to numbers that are expressed using a decimal point. This is a way to represent numbers that are not whole and are divided into tenths, hundredths, thousandths, and so on.
This is opposed to fractions or percentages. For example, the number 7.75 is in decimal form because it has a decimal point separating the whole number 7 and the fractional part, which is 75.
Decimal form is particularly useful because it allows us to see exactly how much of a whole number there is. It is widely used in everyday transactions, like money, where you might see prices written as $3.99 or $7.75, where the numbers after the decimal point represent cents.

Basics of Multiplication

**Multiplication** is one of the four basic arithmetic operations, alongside addition, subtraction, and division. It involves the combining of equal groups. For instance, saying 4 multiplied by 3 means you have three groups of 4.
In mathematical notation, multiplication is often represented by an '×' or sometimes by a dot (·). In the given problem, we are asked to multiply the decimal number 7.75 by the expression \(10^{0}\), meaning that we are essentially taking one group of 7.75.
When multiplying by 1, the outcome remains unchanged, thus confirming that multiplication by unity (1) is uncomplicated and direct. This is foundational, as it serves as the identity element of multiplication, keeping the number the same.

Exploring Exponents

An **exponent** is a number that indicates how many times a base number is multiplied by itself. For example, in the expression \(3^4\), the number 4 is the exponent, and it tells you to multiply the base 3 by itself three more times: \(3 \times 3 \times 3 \times 3 = 81\).
Exponents are a shorthand way to express repeated multiplication. They are closely related to powers and are used in scientific notation to manage very large or small numbers efficiently. In the exercise, \(10^{0}\) utilizes an exponent, where 10 is the base and 0 is the exponent.
Understanding exponents helps in simplifying expressions and solving equations that involve scientific notation, such as the problem given.

The Power of Zero in Exponents

The **power of zero** is a crucial concept in understanding exponents. When any non-zero base is raised to the power of 0, the result is always 1.
This may seem counterintuitive at first; however, it can be understood as following the rules of exponentiation. Consider that \(a^n/a^n = 1\); when you subtract the exponents \( n-n=0 \), you get \(a^0 = 1\). Hence, \(10^0 = 1\).
In the given exercise, understanding that \(10^0 = 1\) simplifies the multiplication process and gives the final result of 7.75 when multiplying \(7.75 \times 10^0\). Recognizing this rule is vital for dealing with expressions involving exponents, ensuring accurate calculations in scientific contexts.