Question

Evaluate the expression. \(8^{2}-17\)

Short answer

The evaluated expression is 47.

Step-by-step solution

Calculate the exponent

Start by calculating \(8^{2}\). The number '2' is the exponent. This means 8 is multiplied by itself 2 times, which is \(8 \times 8 = 64\).

Subtract

After finding the result of the exponentiation operation, subtract 17 from the result: \(64 - 17 = 47\).

Key concepts

Powers and Exponents

Understanding how powers and exponents work is fundamental in evaluating mathematical expressions. An exponent, often called a 'power', represents the number of times a base number is multiplied by itself. For example, when we see an expression like \(8^2\), the number 8 is the base, and 2 is the exponent. It's telling us to multiply 8 by itself one additional time (since 8 is already one instance of itself), resulting in \(8 \times 8 = 64\).

Remember, the base is the big number we see, and the exponent is the little number up high. A handy tip to recall is that any number raised to the power of 1 is itself, and any number raised to the power of 0 is 1 (except for 0 itself, which is a special case). Moreover, when dealing with whole number exponents, the process will always yield a positive result when the base is positive.

Order of Operations

The order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction), guides us in solving expressions in a structured way to avoid confusion and to ensure that everyone solves expressions in the same way, coming to the same result.

According to this rule, calculations involving exponents are completed second, taking precedence after parentheses but before multiplication, division, addition, and subtraction. It’s why, in the original exercise, we handle the exponent \(8^2\) before proceeding to the subtraction of 17. An essential tip here is always to tackle the expression in the correct order to get the correct answer.

Simplifying Expressions

Simplifying expressions is a way to make complex mathematical expressions more understandable and easier to work with. The goal is to condense the expression into its simplest form without changing its value. For the example given \(8^{2}-17\), we simplified it by first dealing with the exponentiation, and then performing the subtraction.

One key strategy when simplifying expressions is to break down the expression into smaller parts and then reassemble the simplified parts back together. It's similar to solving a puzzle by tackling individual pieces first before putting them all together. Additionally, combining like terms and using the distributive property are common methods used in simplification. It's important to remember that you can only simplify expressions; you cannot change their values or the sequence dictated by the order of operations.