Problem 67 - Elementary Algebra

Question

Use the order of operations to simplify the quantities for the following problems. $$ \frac{6^{3}-2 \cdot 10^{2}}{2^{2}}+\frac{18\left(2^{3}+7^{2}\right)}{2(19)-3^{3}} $$

Step-by-Step Solution

Answer

Question: Simplify the given expression, and find its value: $$ \frac{6^{3}-2 \cdot 10^{2}}{2^{2}}+\frac{18\left(2^{3}+7^{2}\right)}{2(19)-3^{3}} $$ Answer: The simplified expression is equal to 97.

Step 1: Solve the first part of the expression

Compute the expression within the first parentheses \(\frac{6^{3}-2 \cdot 10^{2}}{2^{2}}\). $$ \frac{6^3 - 2 \cdot 10^2}{2^2} = \frac{6^3 - 2 \cdot 100}{4} $$ Calculate the exponent and multiplication: $$ \frac{216 - 200}{4} $$ Now, subtract and divide: $$ \frac{16}{4} = 4 $$

Step 2: Solve the second part of the expression

Compute the expression within the second parentheses \(\frac{18\left(2^{3}+7^{2}\right)}{2(19)-3^{3}}\). $$ \frac{18(2^3 + 7^2)}{2(19) - 3^3} = \frac{18(8 + 49)}{38 - 27} $$ Calculate the exponent, addition, and subtraction: $$ \frac{18(57)}{11} $$ Now, multiply and simplify: $$ \frac{1026}{11} = 93 $$

Step 3: Combine the results of both parts

Now that we have the results for both parts of the expression, we will add them to get the final answer. $$ 4 + 93 = 97 $$ Thus, the simplified expression is equal to 97.

Key Concepts

Simplifying Expressions

Simplifying expressions is a foundational skill in algebra, which involves reducing a complex mathematical sentence into a simpler or more compact form without changing its value. This process makes use of the order of operations—often memorized by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction)—to ensure that all mathematicians reach the same result when simplifying an expression.

For instance, in the given exercise, the expression \( \frac{6^{3}-2 \cdot 10^{2}}{2^{2}} \) must be simplified step by step. Here's a deeper insight:

• Exponents are calculated first: \( 6^{3} \) and \( 10^{2} \).
• Following PEMDAS, the multiplication by 2 is done next (\( 2 \cdot 100 \) in this case).
• Subtract the results of these operations.
• Finally, divide by \( 2^{2} \) (or 4), since division comes after the other operations.

This process ensures that every term is accounted for accurately, following the correct order to maintain the integrity of the original expression.

Exponentiation

Exponentiation is an operation involving two numbers: the base and the exponent. The base is the number being multiplied by itself, and the exponent tells us how many times the base is used as a factor. For instance, \( 6^{3} \) means that 6 is multiplied by itself 3 times: \( 6 \times 6 \times 6 \).

It's crucial to remember that the rules of exponentiation include:

• Any number raised to the power of 1 is itself.
• A number raised to the power of 0 is 1 (except for 0).
• The order in which you perform exponentiation matters, as it is not commutative.

In the context of the exercise, exponentiation is one of the first steps to be performed. For the second part of the expression, \( 2^{3} \) and \( 7^{2} \) are calculated before proceeding with multiplication or addition. Understanding and correctly applying exponentiation ensures we start simplification with the right values.

Algebraic Fractions

Algebraic fractions are fractions that include algebraic expressions in the numerator, the denominator, or both. When simplifying algebraic fractions, the goal is to make them as straightforward as possible. This may involve performing operations such as addition, subtraction, multiplication, and division, as well as simplifying any complex expressions.

For example, in the second part of the given problem, we simplify the algebraic fraction \( \frac{18(2^{3} + 7^{2})}{2(19) - 3^{3}} \) by first addressing the numerator and the denominator independently-simplifying each using the order of operations. Once the numerator and denominator are fully simplified, we then divide them to get the simplest form of the fraction, which is a step towards solving the entire expression.

Remember that when simplifying algebraic fractions:

• Always perform the operations inside the parentheses first.
• Apply the rules of exponents before moving to other operations.
• Reduce the fraction to its simplest form, if possible, by dividing both the numerator and the denominator by their greatest common divisor.

Mastering algebraic fractions is essential, as they commonly appear in various fields, including higher-level mathematics, physics, and engineering.