Problem 44 - Algebra 1

Question

Evaluate the expression for the given values of the variables. $$ c^{3}+d \text { when } c=4 \text { and } d=16 $$

Step-by-Step Solution

Answer

The value of the expression \(c^{3}+d\) when \(c=4\) and \(d=16\) is 80.

Step 1: Identify the given values

In this problem, the given values are \(c = 4\) and \(d = 16\).

Step 2: Substitute the values

The next step is to substitute these values into the given expression, which is \(c^{3}+d\). This will become: \((4)^3 + 16\).

Step 3: Simplify

The next and final step is to simplify the expression. The value of \(4^3\) is 64. So the expression becomes \(64 + 16\).

Step 4: Solve

Adding 64 and 16 together gives a final value of 80.

Key Concepts

Substitution in Algebra

Substitution is a fundamental concept in algebra where we replace variables, like 'c' and 'd' in our exercise, with their numerical values. It's like a placeholder in a recipe where you swap 'sugar' for '1 cup of sugar,' — specific and ready for the next step.

In the context of our exercise, we have two variables to deal with: 'c' and 'd'. They have been given particular values, with 'c' equal to 4 and 'd' equal to 16. Substitution, in this case, means that wherever we see 'c', we imagine it replaced with 4, and 'd' replaced with 16, transforming the abstract equation into something more concrete we can compute with ease.

Simplifying Algebraic Expressions

Algebra often presents us with complex expressions, but simplification turns them into a form that is far more manageable. Think of it like decluttering a room so you can easily find your way around. In algebraic terms, this means combining like terms, applying distributive laws, and doing arithmetic operations to make the expressions cleaner and more straightforward.

For instance, in the expression \(c^{3}+d\), once we've substituted 'c' and 'd' with 4 and 16, respectively, the expression simplifies down as we calculate the value of \(4^3\) and add it to 16. This process of executing operations step by step, simplifies the original expression to a single number, making it easily digestible.

Exponents in Algebra

Exponents might seem intimidating, but they are just a shorthand notation in algebra for repeated multiplication. If you have \(c^3\), this simply means \(c \times c \times c\). It's a more compact way to express multiplication that would otherwise take up too much space and time to write out.

In our exercise, when we compute \(4^3\), we're actually calculating \(4 \times 4 \times 4\), which equals 64. It's like saying, 'We need four cups of flour to make these cookies, and we're making three batches.' That would give us 12 cups of flour. The base number is what we multiply, and the exponent tells us how many times we use it in the multiplication.