Problem 81 - Algebra 1: Concepts and Skills

Question

Evaluate the expression. Then simplify the answer. $$ \frac{3 \cdot 7+9}{2^{4}+5-11} $$

Step-by-Step Solution

Answer

The simplified answer to the given expression is 3.

Step 1: Evaluate the numerator

The numerator of the fraction contains a multiplication and an addition operation. According to BODMAS rule, multiplication should be performed first, therefore calculate \(3 \cdot 7\) to get 21. Then add 9 to this result. Thus, the numerator \(3 \cdot 7 + 9 = 21 + 9 = 30\).

Step 2: Evaluate the denominator

In the denominator, there’s an exponentiation, addition, and subtraction operation. According to the BODMAS rule, the exponentiation should be done first. Calculate \(2^{4}\) to get 16, then subtract 11 from the sum of 16 and 5. So, the denominator \(2^{4} + 5 - 11 = 16 + 5 - 11 = 10\).

Step 3: Simplify the fraction

Now you have a fraction \(30/10\). To simplify this fraction to its lowest terms, divide both the numerator and the denominator by their Greatest Common Divisor (GCD), which is 10. Hence, the fraction simplifies to \(3/1 = 3\).

Key Concepts

BODMAS rule

BODMAS stands for Brackets, Orders (i.e., powers and square roots, etc.), Division and Multiplication, Addition and Subtraction. These rules determine the order of operations in mathematical expressions so that everyone simplifies them in the same way.
To put it simply:

• First, calculate anything in brackets or parentheses.
• Second, evaluate any orders, like exponents or square roots.
• Next, perform all division and multiplication from left to right.
• Finally, do any addition and subtraction from left to right.

By following these rules, you can avoid any confusion and inconsistency in mathematical calculations. For example, in our given expression, we first handle the multiplication before moving on to the addition in the numerator, because multiplication comes before addition in BODMAS. In the denominator, exponents take precedence, so we compute them first before proceeding with other operations.

Numerator and Denominator

In any fraction, the numerator is the number above the fraction line, and the denominator is the number below it. The numerator indicates how many parts of the whole we are considering, while the denominator tells us into how many parts the whole has been divided.
To understand with our example:

• The numerator is \(3 \cdot 7 + 9\). Here, following the BODMAS rule, we first resolve the multiplication and then the addition.
• The denominator is \(2^{4} + 5 - 11\). Here, the exponent operation must be calculated first according to BODMAS, followed by the remaining addition and subtraction.

Once both the numerator and the denominator are simplified separately, the entire fraction can be simplified by dividing these two results. This step is crucial for getting the simplest or most reduced form of a fraction, making it easier to work with.

Exponents in Algebra

Exponents, sometimes called powers, are used in algebra to simplify expressions by repeatedly multiplying a number by itself. They are an essential part of various operations and calculations. An exponent can be identified by a small number written above and to the right of a base number, indicating how many times the base number should be multiplied by itself.
For example, in our exercise:

• The term \(2^{4}\) means that the number 2 is multiplied by itself 4 times: \(2 \times 2 \times 2 \times 2\), which equals 16.

Recognizing exponents in expressions lets us properly use the BODMAS rule, prioritizing their evaluation over addition or subtraction. Mastering exponents can make complex algebraic problems much more approachable and less daunting.