In Exercises 51–58, solve each equation. Express the solution in scientific notation.
(2X10⁻⁵)x = 1.2X10⁹
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Step 1: Start by isolating the variable x. To do this, divide both sides of the equation by the coefficient of x, which is 2 \times 10^{-5}.
Step 2: The equation becomes x = \frac{1.2 \times 10^9}{2 \times 10^{-5}}.
Step 3: Simplify the expression by dividing the coefficients: \frac{1.2}{2} = 0.6.
Step 4: Apply the properties of exponents to simplify the powers of 10: \frac{10^9}{10^{-5}} = 10^{9 - (-5)} = 10^{9 + 5} = 10^{14}.
Step 5: Combine the simplified coefficient and power of 10 to express the solution in scientific notation: x = 0.6 \times 10^{14}.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Scientific Notation
Scientific notation is a way of expressing numbers that are too large or too small in a compact form. It is written as a product of a number between 1 and 10 and a power of ten. For example, 1.2 x 10⁹ represents 1.2 billion. Understanding how to convert between standard form and scientific notation is essential for solving equations involving such numbers.
Solving exponential equations involves isolating the variable, often by manipulating the equation to express it in a simpler form. In this case, you would divide both sides of the equation by the coefficient (2 x 10⁻⁵) to solve for x. Mastery of algebraic manipulation and properties of exponents is crucial for finding the correct solution.
The properties of exponents are rules that govern how to handle mathematical operations involving powers. Key properties include the product of powers, quotient of powers, and power of a power. For instance, when multiplying numbers in scientific notation, you add the exponents of the base 10. Understanding these properties is vital for simplifying expressions and solving equations effectively.