Problem 39 - Algebra 1

Question

EVALUATING EXPRESSIONS Evaluate the expression without using a calculator. Write the result in scientific notation and in decimal form. $$ \left(6 \times 10^{5}\right) \cdot\left(2.5 \times 10^{-1}\right) $$

Step-by-Step Solution

Answer

The solution to the expression is \(15 * 10^{4}\) in scientific notation and 150000 in decimal form.

Step 1: Multiply Coefficients

Multiply the coefficients of the terms, which are 6 and 2.5 in this case. The calculation will be \(6 * 2.5 = 15\).

Step 2: Combine the Powers of 10

Combine the powers of 10 by adding the exponents, applying the rule \(10^{n+m}\). The powers are 5 and -1, so the calculation will be \(10^{5 + -1} = 10^{4}\).

Step 3: Put Together the Result in Scientific Notation

Combine the result from step 1 and step 2 to write the answer in scientific notation. The product of the coefficients becomes the new coefficient, and the sum of the exponents becomes the new exponent. Therefore, the solution in scientific notation is \(15 * 10^{4}\).

Step 4: Convert to Decimal Form

To convert \(15 * 10^{4}\) into decimal form, move the decimal in 15 four places to the right (because of \(10^4\)), resulting in \(150000\).

Key Concepts

Understanding Scientific Notation

Scientific notation helps simplify large or small numbers, making calculations more manageable. In this system, numbers are expressed as the product of a coefficient and a power of ten. The format is:

• Coefficient: A number between 1 and 10. E.g. in the expression \(6 \times 10^5\), the coefficient is 6.
• Power of Ten: Indicates how many times to multiply the coefficient by 10. Positive exponents mean a large number, while negatives imply a small number. For example, \(10^5\) means multiplying the coefficient by 10, five times.

Writing numbers in this way makes it easy to compare their sizes and to perform multiplication and division. This is commonly used in scientific calculations, which often involve very large or very small numbers.

Basics of Multiplication of Exponents

Multiplying numbers with exponents involves special rules. If the base is the same, as with the powers of ten, you can add the exponents together. This rule is expressed mathematically as:

• \(a^m \times a^n = a^{m+n}\)

For example, with \(10^5\) and \(10^{-1}\), you add the exponents:

• \(5 + (-1) = 4\)

So, \(10^5 \times 10^{-1} = 10^4\). This simplifies calculations by reducing it to a single power of ten, thus making arithmetic operations easier to handle and understand.

Converting to Decimal Form

Once you've evaluated the expression in scientific notation, converting it to decimal form involves shifting the decimal point. The exponent on ten tells you how many places to move the decimal point:

• Positive Exponent: Move the decimal to the right. The number of places equals the exponent value.
• Negative Exponent: Move the decimal to the left. Again, the number of places equals the absolute value of the exponent.

For instance, the scientific notation \(15 \times 10^4\) translates to decimal form by moving the decimal 4 places to the right, resulting in \(150000\). This process effectively expands the expression, making it easier to interpret in real-world applications.