Problem 89 - Elementary and Intermediate Algebra

Question

Evaluate each expression. $$ 8-3\left[5^{2}-(7-3)^{2}\right] $$

Step-by-Step Solution

Answer

The expression evaluates to -19.

Step 1: Solve inside the parentheses

Focus on the expression inside the parentheses first: \((7-3)\).Calculate \(7-3\) which equals \(4\).

Step 2: Calculate the exponent inside the brackets

Take the result from Step 1 and use it in the exponent within the brackets:\(4^2 = 16.\)

Step 3: Evaluate 5 squared

Now evaluate \(5^2\), which equals \(25\).

Step 4: Solve inside the brackets

Substitute the results from the above steps back into the expression:\(3\left[25 - 16\right]\).Calculate \(25 - 16\), which equals \(9\).

Step 5: Multiply by the factor outside the brackets

Multiply the result from Step 4 by \(3\):\(3 \times 9 = 27\).

Step 6: Subtract the result from the original number

Finally, subtract the result from Step 5 from \(8\): \(8 - 27 = -19.\)

Key Concepts

Exponents

Exponents are a crucial part of mathematics, often appearing in various equations where numbers are multiplied by themselves for a specified number of times. When dealing with an exponent, the base number is raised to the power indicated by the exponent. For example, in the expression \(5^2\), 5 is the base and 2 is the exponent, meaning that 5 should be multiplied by itself once, giving us 25.
To fully grasp exponents, remember:

• The exponent indicates how many times the base number is used in multiplication.
• An exponent of 2 is called "squared" while an exponent of 3 is "cubed".
• Any number to the power of 1 is itself, and any number to the power of 0 is 1.

Working with exponents requires careful calculation, making sure each step is carefully assessed before moving to the next part of the problem.

Arithmetic Operations

Arithmetic operations form the bedrock of all mathematical computation, including addition, subtraction, multiplication, and division. These operations follow a specific order, often remembered by the acronym PEMDAS: Parentheses, Exponents, Multiplication, Division, Addition, Subtraction.
In the given exercise, after dealing with exponents, we handle other arithmetic operations. Start by ensuring

• Multiplication and division are performed from left to right once you've simplified any expressions within parentheses and calculated exponents.
• Only after these steps are addition and subtraction addressed, again from left to right.

Understanding and applying this order accurately is necessary to arrive at the correct solution.

Parentheses

Working with parentheses is a key component in managing complex expressions. Parentheses indicate that the calculations within them should be performed first, prior to any other operations.The expression \((7-3)\) in the problem was solved first, yielding 4. This result was then used in the exponent calculation as follows:

• Always resolve expressions within parentheses first as per the order of operations, ensuring that the solutions inside them are correct before applying other mathematical rules like exponents or outside multipliers.
• If multiple sets of parentheses are present, work from the innermost set outwards.
• In cases where brackets are also present, follow the same approach where you resolve parentheses before moving on to work within other brackets.

By sticking to these rules, calculations within parentheses simplify complex mathematical expressions step by step, making the overarching problem easier to tackle.