Problem 65 - College Algebra and Calculus: An Applied Approach

Question

Evaluate each expression without using a calculator.(a) \(\left(1.2 \times 10^{7}\right)\left(5 \times 10^{-3}\right)\) (b) \(\frac{6.0 \times 10^{8}}{3.0 \times 10^{-3}}\)

Step-by-Step Solution

Answer

The solutions for the two expressions are respectively \(6 \times 10^{4}\) and \(2.0 \times 10^{11}\).

Step 1: Analyze the First Expression

The first expression is in multiplication format, which is \(1.2 \times 10^{7}\) times \(5 \times 10^{-3}\). First, multiply the two numbers 1.2 and 5, then add the powers.

Step 2: Solve the First Expression

Multiply the numbers 1.2 and 5 will get 6. For the exponent part, add the exponents together: \(10^{7} \times 10^{-3}\) equals to \(10^{7-3}\), which equals to \(10^{4}\). Therefore, the result of the first expression is \(6 \times 10^{4}\).

Step 3: Analyze the Second Expression

The second expression is in division format, which is \(\frac{6.0 \times 10^{8}}{3.0 \times 10^{-3}}\). First, divide the two numbers 6.0 and 3.0, then subtract the powers.

Step 4: Solve the Second Expression

Divide the numbers 6.0 by 3.0 will get 2.0. For the exponent part, subtract the bottom exponent from the top exponent: \(\frac{10^{8}}{10^{-3}}\) equal to \(10^{8 - (-3)}\), which equals to \(10^{11}\). Therefore, the result of the second expression is \(2.0 \times 10^{11}\).

Key Concepts

Exponent Laws

Exponent laws are the rules that govern the operations performed on exponents when they are involved in multiplication, division, and other mathematical operations. The two most common laws used in scientific notation are the Product of Powers and the Quotient of Powers.

• The Product of Powers rule states that when you multiply two exponents with the same base, you add the exponents: \( a^m \times a^n = a^{m+n} \).
• The Quotient of Powers rule states that when you divide two exponents with the same base, you subtract the exponents of the divisor from the exponents of the dividend: \( \frac{a^m}{a^n} = a^{m-n} \).

Understanding these rules is essential when working with scientific notation, as it often involves multiplying and dividing numbers with exponents.

Scientific Notation Multiplication

Scientific notation makes it easier to handle very large or very small numbers by expressing them as a product of a base number (between 1 and 10) and a power of 10. To multiply numbers in scientific notation, you multiply the base numbers and apply the exponent laws to the powers of 10.

For example, if you want to multiply \(1.2 \times 10^{7}\) and \(5 \times 10^{-3}\), you would multiply 1.2 by 5 to get 6, and then add the exponents (7 and -3) to get 4. The final result would be \(6 \times 10^{4}\). It's a straightforward process that simplifies calculations involving very large or very small numbers.

Scientific Notation Division

Similarly, when dividing numbers in scientific notation, divide the base numbers and use the exponent laws for the powers of 10.

Consider the expression \(\frac{6.0 \times 10^{8}}{3.0 \times 10^{-3}}\). You would divide 6.0 by 3.0 to get 2.0, and then subtract the denominator's exponent (-3) from the numerator's exponent (8) according to the exponent laws, thus \(8 - (-3) = 11\). The result is \(2.0 \times 10^{11}\), reflecting the precision of the original numbers while simplifying the operation significantly.

Algebraic Expressions

In addition to exponent laws, understanding how to simplify algebraic expressions is crucial. An algebraic expression is a mathematical phrase that can contain numbers, variables, and operations. The manipulation of these expressions is governed by the rules of arithmetic.

In scientific notation, the algebraic expressions become more manageable due to the structure of having a coefficient multiplied by a power of ten. This framework provides a systematic method for multiplication and division, as illustrated in the previous examples, and it's a fundamental aspect of algebra that enables students to simplify and solve various scientific and mathematical problems.