91Ó°ÊÓ

Exponents are a fundamental concept in mathematics that describe how many times a number, known as the base, is multiplied by itself. For example, in the expression \(2^x\), the number 2 is the base, and \(x\) is the exponent. When we say \(2^3\), it means 2 is multiplied by itself three times: \(2 \times 2 \times 2 = 8\). This concept is crucial when solving exponential equations.Understanding exponents helps solve equations where the variable is part of the exponent. Such equations often require expressing all terms with the same base, simplifying, and then equating the exponents. In the exercise above, recognizing \(8\) as \(2^3\) allows us to express all terms as powers of 2, enabling the next steps in solving the equation.

An equation is a mathematical statement that shows the equality between two expressions. The goal in solving equations is to find the value of the variables that make the equation true. In algebra, especially with exponential equations, it's common to manipulate the equation to isolate the variable.In our original exercise, the equation is given as:\[ \frac{1}{2^x} = 8 \cdot 2^x \]For exponential equations like this, the strategy is often to express both sides with a common base, which may involve converting numbers like 8 to its equivalent power of 2. Rewriting all terms in terms of the same base as:\[ \frac{1}{2^x} = 2^3 \cdot 2^x \]allows us to simplify and equate the exponents, moving forward to find the solution for \(x\).

Algebraic manipulation involves using mathematical techniques and operations to simplify and solve equations. When handling exponential equations, this often includes simplifying expressions with the same base, combining like terms, and performing operations to isolate the variable.In the exercise, once we have the equation:\[ 2^{-x} = 2^{3+x} \]we use algebraic manipulation to set the exponents equal:\[ -x = 3 + x \]By adding \(x\) to both sides, we move terms involving the variable to one side of the equation:\[ 0 = 3 + 2x \]Next, we solve for \(x\) by isolating it. Subtract 3 from both sides and then divide by 2:\[ x = -\frac{3}{2} \]These steps demonstrate how algebraic manipulation seamlessly connects with the principles of exponents and equations, allowing us to solve for unknown values efficiently.