Problem 53 - Algebra 2

Question

Mental Math Solve each equation. $$ 2^{x}=\frac{1}{2} $$

Step-by-Step Solution

Answer

The solution to the equation is \(x=-1\).

Step 1: Identify the base of the exponent

The exponential equation is given in the form \(2^{x}=\frac{1}{2}\). Here, the base of the exponent on the left side is 2.

Step 2: Rewrite the right side of the equation

The fraction \(\frac{1}{2}\) can also be written as \(2^{-1}\) because any number n to the power of -1 is equal to \(\frac{1}{n}\). So we rewrite the equation as \(2^x=2^{-1}\).

Step 3: Compare the powers

Since the bases are the same on both sides of the equation, we can equate the powers. Thus we have \(x=-1\).

Key Concepts

Exponents

Exponents are a way to express repeated multiplication. In the expression \(a^n\), \(a\) is the base and \(n\) is the exponent. This is read as "\(a\) to the power of \(n\)." For example, \(2^3\) is equal to \(2 \times 2 \times 2 = 8\). Exponents help in simplifying long multiplication processes.
Key exponent rules include:

• Multiplying like bases: Add the exponents: \(a^m \times a^n = a^{m+n}\).
• Dividing like bases: Subtract the exponents: \(a^m / a^n = a^{m-n}\).
• Power of a power: Multiply the exponents: \((a^m)^n = a^{m \times n}\).

Using these rules is crucial when solving equations involving exponents, as they allow us to manipulate expressions in a consistent way.

Mental Math

Mental math is all about performing calculations in your head, without using a calculator or pen and paper. In the context of solving exponential equations, mental math helps in quickly identifying equivalent expressions.

For example:

• Recognizing that \(\frac{1}{2}\) can be written as \(2^{-1}\) makes it easier to solve the equation \(2^x = \frac{1}{2}\) by simply equating the exponents mentally.
• Familiarity with powers of small numbers like 2 or 10 can speed up calculations and verification of solutions.

Strengthening mental math skills requires practice but aids significantly in problem-solving, especially when dealing with common bases and exponents.

Equation Solving

Equation solving is the process of finding unknown variables that satisfy an equation. For exponential equations, particularly, the goal is often to rewrite or simplify expressions until a solution for the variable is found.

When given \(2^x = 2^{-1}\), we compare the exponents once both sides of the equation have the same base. Here, both sides already have a base of 2, so we set \(x = -1\).

• This method of "equating the exponents" is efficient because whenever the bases are identical, the exponents must be equal for the expressions to be equal.
• Equation solving might include additional steps involving logarithms if the bases cannot be directly compared.

Understanding base manipulation and exponent rules will make solving these equations more intuitive.

Negative Exponents

Negative exponents are used to indicate reciprocals. They follow a simple principle: for any non-zero base \(a\), \(a^{-n} = \frac{1}{a^n}\). This turns problems involving division into simpler multiplication ones using positive exponents.

In our exercise, \(2^{-1}\) is used, which translates directly to \(\frac{1}{2}\). Understanding this conversion allows for easier manipulation of equations, as observed in the solution.
Considerations:

• Negative exponents flip the base into the denominator of a fraction.
• This conversion is useful for solving equations and simplifying complex expressions.

Being comfortable with moving between negative exponents and fractions is a crucial skill in algebra.