Problem 18 - Algebra 1

Question

DECIMAL FORM Rewrite in decimal form. $$ 7.75 \times 10^{0} $$

Step-by-Step Solution

Answer

The decimal form of \(7.75 \times 10^0\) is 7.75.

Step 1: Understand the scientific notation

In this expression, 7.75 is the coefficient and 0 is the exponent. In scientific notation, this is written as \(a \times 10^n\).

Step 2: Apply the exponent rule

Any number raised to the zeroth power is equal to one. Therefore, \(10^0 = 1\).

Step 3: Convert the scientific notation to decimal form

Now, to convert the expression into decimal form, simply multiply the coefficient 7.75 by \(10^0 = 1\). Therefore, \(7.75 \times 10^0 = 7.75 \times 1 = 7.75\).

Key Concepts

Scientific Notation

Scientific notation is a method used to express extremely large or small numbers in a concise form. It is composed of two parts: a coefficient and an exponent. The coefficient is a number between 1 and 10, and the exponent indicates how many times the coefficient should be multiplied by ten. For example, the scientific notation of the number 500 is written as \(5 \times 10^2\), which means the coefficient 5 is multiplied by 10 raised to the power of 2.

When numbers are written in scientific notation, it is easier to handle complex calculations, especially in fields like physics and chemistry. It simplifies the understanding of magnitude and scales since it can neatly represent values that would otherwise require many zeros to write out in decimal form.

Exponent Rules

Exponents are shorthand for repeated multiplication, and certain rules make working with exponents simpler. Here are crucial exponent rules to remember:

• The Product Rule: \(a^m \times a^n = a^{m+n}\), which means when multiplying two powers with the same base, add the exponents.
• The Quotient Rule: \(\frac{a^m}{a^n} = a^{m-n}\), which tells us to subtract the exponents when dividing two powers with the same base.
• The Power of a Power Rule: \((a^m)^n = a^{m \times n}\), this states that when raising a power to another power, multiply the exponents.
• The Zero Exponent Rule: Any base (except zero) raised to the power of zero is always one, so \(a^0 = 1\).

These rules are foundational in algebra and help in simplifying and solving algebraic expressions involving exponents.

Decimal Conversion

Converting scientific notation to decimal form involves using the exponent part of the expression to determine where to place the decimal point. For exponents that are positive, you move the decimal point to the right, each move corresponding to a factor of ten. Conversely, for negative exponents, you move the decimal point to the left, indicating division by ten for each step.

For example, to convert \(3.2 \times 10^3\) to decimal form, move the decimal point three places to the right, which gives you 3200. It is also important to note that if the exponent is zero, as in the exercise example \(7.75 \times 10^0\), this signifies that the number is already in decimal form because any number to the power of zero equals one.

Algebraic Expressions

Algebraic expressions represent quantities in a general form and include numbers, variables, and arithmetic operations. They can range from simple forms like \(2x+3\) to more complex forms with multiple terms and exponents. Simplifying algebraic expressions involves combining like terms and applying exponent rules.

An understanding of how to properly apply mathematical laws to manipulate and simplify these expressions is critical in solving algebra problems. For instance, understanding that the zeroth power of any base (other than zero) results in one simplifies expressions like \(x^0\), which equals one. This simplification often becomes the critical step in unlocking the simplest form of an algebraic expression and solving equations.