Problem 43 - Algebra 1: Concepts and Skills

Question

Evaluate the variable expression when c 4 and d 5. $$ \left(d^{2}\right)-\left(c^{2}\right) $$

Step-by-Step Solution

Answer

The value of the expression when c = 4 and d = 5 is 9.

Step 1: Identify the Variables

In the given expression, identify the variables. Here, they are c and d.

Step 2: Substitute values

Substitute the given values into the equation. In this case, substitute c with 4 and d with 5. Now the expression becomes: \( (5^2) - (4^2) \).

Step 3: Simplify the expression

Calculate the squares, which gives \( 25 - 16 \).

Step 4: Perform the subtraction

Finally, subtract 16 from 25, which yields the result 9.

Key Concepts

Substitution in Algebra

Substitution is a fundamental concept in algebra that involves replacing variables with their corresponding numerical values. It's like a 'switcheroo'—wherever you see the variable, you swap it out for a number. For instance, if you're given an equation and told that c=4 and d=5, you'd replace every instance of c with 4 and every d with 5. This step is pivotal for solving the equation, as it makes the 'unknowns' known and the expression numerically solvable.

Here's a quick tip: Always use parentheses when substituting, just like in \( (d^2) - (c^2) \) where \( d \) and \( c \) are replaced with \( (5^2) \) and \( (4^2) \) respectively. This prevents mistakes often made due to overlooking operations like exponentiation or multiplication that should apply to the substituted number as a whole.

Simplifying Algebraic Expressions

Simplifying algebraic expressions is like cleaning your room; it's all about making things neater and more manageable. When confronted with an algebraic expression, your goal is to combine like terms and perform any operations to reduce it to its simplest form. Like our exercise above, you want to make sure you carry out any exponentiation, multiplication, addition, or subtraction correctly.

Here's how you do it step-by-step:

• Address the exponentiation first, according to 'order of operations' rules (remember BODMAS/BIDMAS?).
• Once your variables are substituted, calculate their exponentiated values.
• Next, look for like terms to combine.
• Last, finish off any remaining operations.

In our example, \( (5^2) - (4^2) \) simplifies down to \( 25 - 16 \) once we've followed these steps.

Basic Algebraic Operations

Algebra isn't all letters and unknowns—it's founded on basic operations you know and love: addition, subtraction, multiplication, and division. These operations are the core tools you'll use to manipulate algebraic expressions. They follow the same rules you've been using since grade school, but now, your challenge lies in applying them to variables and constants alike.

When you get an expression like \( (d^2) - (c^2) \) and your variables are nicely replaced by numbers, it's game time for these operations. Calculate powers first (exponentiation), then move on to the rest. In our example, after calculating the squares of 5 and 4, the last basic operation you'll perform is subtraction: \( 25 - 16 \) which equals \( 9 \).

Exponentiation

Exponentiation might seem fancy, but it's just repeated multiplication. When you see \( 5^2 \), it means \( 5 \) times \( 5 \) — that's \( 25 \). And \( 4^2 \) is \( 4 \) times \( 4 \) — which makes \( 16 \). In algebra, dealing with exponents often comes before any other operation (except parentheses!), so always handle these first.

Keep an eye out for exponents; they can seriously change your results. If instead of squaring, you simply multiplied by 2, you'd get the wrong answer. When variables with exponents are involved, correctly evaluate these variables after substitution, like we did with \( c \) and \( d \) in the given problem.