Problem 95 - Elementary and Intermediate Algebra

Question

Evaluate each expression. See Example 10. $$ \left(27 c^{2}-4 d^{2}\right)^{3} \text { for } c=\frac{1}{3} \text { and } d=\frac{1}{2} $$

Step-by-Step Solution

Answer

The evaluated expression is 8.

Step 1: Substitute the Values

First, substitute the given values of \( c \) and \( d \) into the expression \( 27c^2 - 4d^2 \). For \( c = \frac{1}{3} \) and \( d = \frac{1}{2} \), substitute these into the expression to get: \[ 27 \left(\frac{1}{3}\right)^2 - 4 \left(\frac{1}{2}\right)^2 \]

Step 2: Calculate \( c^2 \) and \( d^2 \)

Calculate the square of \( c \) and \( d \). Thus, \( c^2 = \left(\frac{1}{3}\right)^2 = \frac{1}{9} \) and \( d^2 = \left(\frac{1}{2}\right)^2 = \frac{1}{4} \).

Step 3: Simplify the Expression

Substitute the squared values back into the expression: \[ 27 \times \frac{1}{9} - 4 \times \frac{1}{4} = 3 - 1 = 2 \]

Step 4: Raise the Result to the Power of 3

Now take the result of \( 2 \) and raise it to the power of 3: \( 2^3 = 8 \).

Key Concepts

Algebraic Expressions

An algebraic expression is a combination of variables, constants, and mathematical operators, such as addition or multiplication.
They can appear quite complex, but they are simply the connection of numbers and symbols to represent a value or quantity. For example, the algebraic expression given here, \(27c^2 - 4d^2\), assigns unknown quantities to the variables \(c\) and \(d\). The key is to understand what each part of the expression is doing:

• **Constants**: These are the fixed numerical values like 27 and 4 in the expression.
• **Variables**: Represented by letters such as \(c\) and \(d\), which can take on various numerical values.
• **Exponents**: These indicate how many times a number (base) is multiplied by itself.

Once you're comfortable with these components, evaluating algebraic expressions becomes much simpler.

Substitution Method

The substitution method is a way to evaluate algebraic expressions by replacing variables with their specific values.
This is often the first step in solving an expression or equation where you have specific information about the variables.In practice, you replace each variable in the expression with its given value.
For example, if you know that \(c = \frac{1}{3}\) and \(d = \frac{1}{2}\), you would substitute directly into the expression:

• Replace \(c\) with \(\frac{1}{3}\)
• Replace \(d\) with \(\frac{1}{2}\)

This step transforms the expression into something entirely numerical and makes it ready for further simplification or calculation.
Substitution is fundamental in making abstract expressions more tangible and easier to evaluate.

Exponents

Exponents are a shorthand notation to denote repeated multiplication of a number by itself.
They are expressed as a number (the base) raised to a power (the exponent). For instance, \((\frac{1}{3})^2\) means multiplying \(\frac{1}{3}\) by itself:

• \( (\frac{1}{3}) \times (\frac{1}{3}) = \frac{1}{9} \)

Working with exponents requires understanding the basic rules:

• **Product Rule**: Multiply like bases by adding exponents.
• **Power Rule**: Raise a power to another power by multiplying the exponents.
• **Quotient Rule**: Divide like bases by subtracting exponents.

In the given problem, exponents help simplify and solve expressions by reducing the need for repetitive multiplication.

Mathematical Operations

Mathematical operations such as addition, subtraction, multiplication, and division are the backbone of simplifying expressions.
They follow a specific order known as BODMAS/BIDMAS (Brackets, Orders (i.e., powers and roots, etc.), Division and Multiplication, Addition and Subtraction).Here’s how you apply these operations:

• **Brackets/Parentheses**: Solve anything inside them first.
• **Orders**: Calculate exponents like \(c^2\) and \(d^2\).
• **Multiplication/Division**: Next, perform any operations according to their positioning from left to right.
• **Addition/Subtraction**: Lastly, perform addition or subtraction as necessary.

Using these operations correctly allows us to simplify the expression step-by-step:
Calculate each power, multiply these results by the coefficients (27 and 4), and finally, perform the subtraction.
Once simplified, any remaining operations like raising a result to a power, as in \(2^3\), should be handled according to these rules.