Question

Evaluate the expression. \(\frac{1}{2} n^{3}\) when \(n=-2\)

Short answer

The evaluated expression is \(-4\).

Step-by-step solution

Substitution

First, substitute the given value of \(n = -2\) into the expression \(\frac{1}{2} n^{3}\). This gives us \(\frac{1}{2} (-2)^{3}\).

Exponentiation

Next, compute the exponentiation part \((-2)^{3}\). This results in \(-8\). So the expression now transforms to \(\frac{1}{2} * -8\).

Multiplication

Finally, calculate the product \(\frac{1}{2} * -8\). The answer is \(-4\).

Key concepts

Exponentiation

In mathematics, exponentiation is an operation that involves raising a number, known as the base, to the power of an exponent. The exponent signifies how many times the base is multiplied by itself. For example, in the expression \(b^n\), \(b\) is the base, and \(n\) is the exponent or power.

Let's apply this to the exercise where we have \(n^3\). This means \(n\) is the base, and it is multiplied by itself three times: \(n \times n \times n\). When \(n=-2\), as in the given expression, \(n^3\) becomes \((-2) \times (-2) \times (-2)\) which equals \(-8\), because a negative number multiplied by itself an odd number of times results in a negative product.

Remember that the sign of the result depends on the exponent: if it's even, the result is positive, but if it's odd—as in our case—the result is negative.

Substitution in Algebra

The technique of substitution in algebra is essentially replacing a variable with its value. This helps in simplifying expressions or solving equations. When we have an expression with a variable, like \(\frac{1}{2} n^{3}\), and we know the value of that variable, we can substitute the variable with its value for further calculation.

In our exercise, we substitute \(n\) with \(n=-2\). With substitution, we go from the abstract \(\frac{1}{2} n^{3}\) to the more concrete \(\frac{1}{2} (-2)^{3}\), which we can compute. Proper substitution is crucial as the next steps of the problem-solving process completely depend on it. Substitution allows for a clearer understanding and often simplifies the steps needed to find a solution.

Multiplication of Fractions

The process of multiplying fractions is straightforward: Simply multiply the numerators (top numbers) together, and then the denominators (bottom numbers) together. The rule is \(\frac{a}{b} \cdot \frac{c}{d} = \frac{a \cdot c}{b \cdot d}\). Unlike when adding or subtracting fractions, you don't need a common denominator.

In the context of our problem, we multiply the fraction \(\frac{1}{2}\) by \(\frac{-8}{1}\)—since any integer can be written as a fraction with 1 as the denominator—giving us \(-4\) because \(\frac{1}{2} \cdot \frac{-8}{1} = \frac{1 \cdot -8}{2 \cdot 1} = \frac{-8}{2}\), which simplifies to \(\frac{-8}{2} = -4\). Multiplication of fractions is often used together with other operations, such as exponentiation and substitution, to evaluate expressions like the one in this exercise.