Problem 10 - Intermediate Algebra

Question

Fill in the blanks. Perform a check to determine whether \(-2\) is a solution of \(5^{2 x+3}=\frac{1}{5}\)

Step-by-Step Solution

Answer

Yes, \(-2\) is a solution because it satisfies the equation.

Step 1: Plug the Value into the Equation

We need to check if \(-2\) is a solution to the equation \(5^{2x+3}=\frac{1}{5}\). Begin by substituting \(-2\) for \(x\) in the exponent: \(5^{2(-2)+3}\).

Step 2: Simplify the Exponent

Calculate the exponent by simplifying it: \(2(-2) + 3 = -4 + 3 = -1\).

Step 3: Evaluate the Power

With the exponent simplified to \(-1\), compute \(5^{-1}\). This equals \(\frac{1}{5^1} = \frac{1}{5}\).

Step 4: Compare with Right Side of the Equation

The result \(\frac{1}{5}\) matches the right side of the equation \(\frac{1}{5}\). This confirms that both sides are equal when \(x = -2\).

Key Concepts

Understanding Exponents

Exponents are a mathematical way of expressing repeated multiplication of the same number. For example, when you see \(5^3\), it is the same as saying \(5 \times 5 \times 5\). The number 5 is the base, and 3 is the exponent, indicating how many times to multiply the base by itself.

Exponents follow specific rules that make calculations manageable:

• Multiplication of Powers: \(a^m \times a^n = a^{m+n}\).
• Division of Powers: \(a^m \div a^n = a^{m-n}\).
• Power of a Power: \((a^m)^n = a^{m \times n}\).
• Zero Exponent: \(a^0 = 1\), assuming \(a e 0\).
• Negative Exponents: \(a^{-n} = \frac{1}{a^n}\).

Negative exponents, like the one in our example \(5^{-1}\), mean that the base appears in the denominator instead of the numerator. Hence, \(5^{-1} = \frac{1}{5}\). Knowing how to handle negative exponents is crucial for solving equations that involve them.

Solving Equations Involving Exponents

In algebra, an equation is a statement that asserts the equality of two expressions. Solving equations with exponents, like the example \(5^{2x+3} = \frac{1}{5}\), involves manipulating the equation to isolate the variable.

• Substitution: To verify if a specific value of \(x\) is a solution, start by substituting that value into the exponents of the equation. For \(x = -2\), substitute into \(5^{2x+3}\) to get \(5^{2(-2)+3}\).
• Simplification: Each arithmetic operation is performed in sequence. For instance, calculate \(2(-2) + 3 = -4 + 3 = -1\).
• Evaluation of Powers: Once the exponent is simplified, evaluate the power, \(5^{-1}\), which equals \(\frac{1}{5}\).

The secret to solving these kinds of equations is careful arithmetic and a strong understanding of how exponents work.

Verification of Solutions

Verification of solutions involves substituting the solution back into the original equation to check for correctness. It ensures that the calculated solution works in the given equation.

• Plugging Back: Substitute the value of \(x\) back into the original equation. In our case, substitute \(x = -2\) to confirm \(5^{2x+3} = \frac{1}{5}\).
• Evaluating Both Sides: Calculate and simplify both sides of the equation to check if they are equal. This step verifies that the left side equals the right side, i.e., \(5^{-1} = \frac{1}{5}\).
• Confirmation: Once both sides of the equation match, you have proven that the value is indeed a solution. This is an essential step because it confirms the correctness of your earlier calculations and assumptions.

Verification is crucial because it assures you that the operations and simplifications you performed were correct, thereby reinforcing your solution.