Problem 67 - Elementary and Intermediate Algebra

Question

Evaluate. $$ 24 \div t^{3}, \text { for } t=-2 $$

Step-by-Step Solution

Answer

The expression evaluates to -3.

Step 1: Substitute the value of t

Substitute the given value of \( t = -2 \) into the expression \( 24 \div t^{3} \). This gives: \( 24 \div (-2)^{3} \).

Step 2: Calculate the exponent

Calculate \( (-2)^{3} \). Since multiplying \( -2 \) by itself three times gives \( -8 \), we have: \( (-2)^{3} = -8 \).

Step 3: Perform the division

Divide 24 by -8: \( 24 \div (-8) \). This results in: \( 24 \div (-8) = -3 \).

Key Concepts

Substitution

Substitution is a fundamental concept in algebra where you replace variables with specific values. We did this in our exercise by substituting the given value of \( t = -2 \) into the expression. This helps us convert an algebraic expression into a simpler numerical problem that we can solve step by step. Always be careful with the signs when doing substitution, especially if the variable is negative.

Exponents

Exponents indicate how many times a number is multiplied by itself. In our exercise, we needed to calculate \( (-2)^3 \). This means \( -2 \times -2 \times -2 \). When you multiply negatives, remember:

• Negative times negative is positive
• Positive times negative is negative

So, in three steps, we get: \( -2 \times -2 = 4 \), and then \( 4 \times -2 = -8 \). Hence, \( (-2)^3 = -8 \).

Division

Division is the process of finding out how many times one number is contained within another. In our exercise, we performed the division \( 24 \div (-8) \).
Remember the rule for dividing numbers with signs:

• Positive divided by negative results in negative.
• Negative divided by positive also results in negative.
• Positive divided by positive or negative divided by negative results in positive.

So, \( 24 \div -8 \) results in \( -3 \). Hence, the final answer is \( -3 \).