Problem 113 - Algebra and Trigonometry

Question

Use the order of operations to simplify each expression. $$\frac{5 \cdot 2-3^{2}}{\left[3^{2}-(-2)\right]^{2}}$$

Step-by-Step Solution

Answer

\( \frac{1}{121} \).

Step 1: Deal with the expression inside the brackets

Firstly, calculate the expressison inside the square brackets. Start by calculating the exponent (or power), which is \(3^2 = 9\), then add \(-(-2)\) which equals to \(2\). The result is \(9+2=11\). So \([3^{2}-(-2)]^{2}\) simplifies to \(11^{2}\).

Step 2: Simplify the numerator

To simplify the numerator \((5 \cdot 2-3^{2})\), start by performing multiplication and then subtraction. So, \(5 \cdot 2 = 10\) and \(3^{2} = 9\). Subtract the \((3^{2})\) from \(10\) to get \(1\). So \(5 \cdot 2-3^{2}\) simplifies to \(1\).

Step 3: Replace the numerator and denominator

Now replace the numerator and denominator with the results from step 1 and step 2. The expression becomes \( \frac{1}{11^2} \).

Step 4: Simplify the expression

Finally, calculate the denominator \(11^{2} = 121\). So, the expression \( \frac{1}{11^2} \) simplifies to \( \frac{1}{121} \).