Problem 25 - Algebra 2

Question

Evaluate each expression if \(a=\frac{2}{5}, b=-3, c=0.5,\) and \(d=6\). \(b^{4}-d\)

Step-by-Step Solution

Answer

The expression evaluates to 75.

Step 1: Substitute the given values into the expression

In the expression \(b^{4}-d\), replace \(b\) with \(-3\) and \(d\) with \(6\). This gives us \((-3)^{4} - 6\).

Step 2: Simplify the power operation

Calculate \((-3)^{4}\). Since raising a negative number to an even power results in a positive number, we have \((-3)^{4} = 3^{4} = 81\).

Step 3: Perform the subtraction

Now that we have the power calculated, substitute back into the expression: \(81 - 6\). Perform the subtraction to get \(81 - 6 = 75\).

Key Concepts

Substitution Method

The substitution method is a powerful tool in mathematics that allows you to solve equations by replacing variables with their given numerical values. This simplifies the expression and makes solving it straightforward. In the context of evaluating expressions, substitution involves:

• Identifying the variables in your equation.
• Looking for the values provided for each variable.
• Replacing each variable in the equation with its corresponding value.

For instance, in the expression \(b^{4} - d\), substitute \(b = -3\) and \(d = 6\). This gives you a new expression to work with: \((-3)^{4} - 6\). By replacing the variables, we make it possible to then proceed with further operations like calculating exponents or performing arithmetic operations.

Negative Numbers

Negative numbers are numbers less than zero and are often represented with a minus sign (-). They can have distinct effects, especially when involved in operations like powers. When raising negative numbers to a certain power, it's important to understand:

• An even exponent will turn a negative number positive, because multiplying two negative numbers results in a positive number.
• An odd exponent will keep the negative number negative.

In our example, raising \(-3\) to the fourth power \((-3)^{4}\) involves understanding that \(-3\) multiplied by itself four times: \((-3) \times (-3) \times (-3) \times (-3)\) results in \(81\). Notice here that because we are using an even exponent \(4\), the result is positive.

Powers of Integers

In mathematics, an exponent or power is a small number placed to the top-right of a base number, indicating how many times the base number is used in multiplication. Powers of integers are straightforward when you know the rules:

• For any integer \(a\) raised to the power of \(n\), it means \(a\) multiplied by itself \(n\) times: \(a^n = a \times a \times \ldots \times a\) (\(n\) times).
• Exponents are particularly helpful in simplifying multiplication of the same number repeated several times.

In our exercise, we evaluate \((-3)^{4}\). After replacing \(-3\) with \(b\), we use the understanding of powers to compute that \((-3)^{4} = 3 \times 3 \times 3 \times 3 = 81\). This shows the power of integers in action, allowing complex operations to be simplified and solved efficiently.