Problem 43 - Algebra 1

Question

EVALUATING EXPRESSIONS Evaluate the expression without using a calculator. Write the result in scientific notation and in decimal form. $$ \frac{6.6 \times 10^{-1}}{1.1 \times 10^{-1}} $$

Step-by-Step Solution

Answer

The answer is \(6.0 \times 10^0\) in scientific notation and 6 in decimal form.

Step 1: Separate Coefficients and Powers of 10

Break down the expression by separating the coefficients from the powers of 10. The expression becomes \((6.6/1.1) \times (10^{-1}/10^{-1})\).

Step 2: Simplify Division

Perform the division of the coefficients and the powers of 10. The division of the coefficients is \(6.6/1.1 = 6\), and for the powers of 10, subtract the exponent in the denominator from the exponent in the numerator, hence \(10^{-1}-(-1) = 10^0\).

Step 3: Write in Scientific Notation

The simplified expression should be written in scientific notation. Remember that any number raised to the power of 0 is 1, hence \(10^0 = 1\). The scientific notation of \((6 \times 1)\) becomes \(6.0 \times 10^0\).

Step 4: Convert to Decimal Form

The decimal form of \(6.0 \times 10^0\) is simply 6 because any number multiplied by 1 remains the same.

Key Concepts

Scientific Notation

Scientific notation makes handling very large or very small numbers much easier. It expresses numbers as the product of two parts: a coefficient and a power of 10. For example, the number \(4,000,000\) can be written as \(4.0 \times 10^6\). This is especially useful in science and engineering, where such numbers are common.

• The coefficient is a decimal number greater than or equal to 1 and less than 10.
• The power of 10 indicates how many times the coefficient should be multiplied by 10.

Scientific notation is not only about convenience. It provides precision as well. For instance, writing 4.0 indicates one significant figure, whereas 4.00 indicates two. Understanding this concept is key to performing calculations like the one in the exercise, where you simplify expressions and express them in a neat manner.

Decimal Form

Decimal form is the standard way of writing numbers using our base-10 numbering system. In decimal form, numbers are written out fully, showing their complete value. For example, \(6.0 \times 10^0\) written in decimal form is simply 6.When converting from scientific notation to decimal form:

• If the exponent is positive, move the decimal point to the right the number of times indicated by the exponent.
• If the exponent is negative, move the decimal point to the left.

In our example, since \(10^0\) equals 1, the number remains unchanged as 6. This process ensures clarity and simplicity, enabling anyone to understand the value of a number at a glance.

Powers of 10

Powers of 10 are pivotal in the concept of scientific notation and are incredibly useful for simplifying calculations. A power of 10 is created by raising the number 10 to an exponent. For example, \(10^3 = 1,000\) and \(10^{-3} = 0.001\). In any expression, exponents follow specific mathematical rules that make calculations more straightforward:

• When multiplying like bases, add the exponents: \(10^a \times 10^b = 10^{a+b}\).
• When dividing, subtract the exponents: \(10^a/10^b = 10^{a-b}\).
• Any number to the power of zero equals one: \(10^0 = 1\).

In the given exercise, the simplified expression \(6.0 \times 10^0\) leverages these rules effectively: the division of \(10^{-1}\) by \(10^{-1}\) results in \(10^0\), simplifying the entire calculation. Understanding powers of 10 is crucial for comprehending and evaluating expressions in scientific notation efficiently.