Problem 45 - Algebra for College Students

Question

Simplify each of the numerical expressions. $$\left[3(-2)^{2}-2(-3)^{2}\right]^{3}$$

Step-by-Step Solution

Answer

The simplified expression is -216.

Step 1: Evaluate the Exponents

First, evaluate the exponents inside the brackets. For \((-2)^{2}\), the base is -2, and the exponent is 2, so \((-2)^{2} = 4\). Similarly, for \((-3)^{2}\), the base is -3, and the exponent is 2, so \((-3)^{2} = 9\).

Step 2: Substitute and Multiply

Substitute the evaluated exponents back into the expression: \(3 \times 4 - 2 \times 9\). Now, perform the multiplications: \(3 \times 4 = 12\) and \(2 \times 9 = 18\).

Step 3: Perform the Subtraction

Subtract the two results from Step 2: \(12 - 18 = -6\).

Step 4: Evaluate the Cubed Term

Now, cube the result from Step 3: \((-6)^{3}\). To do this, multiply -6 by itself three times: \(-6 \times -6 = 36\) and then \(36 \times -6 = -216\).

Key Concepts

Evaluating Exponents

Exponents are powerful mathematical tools that quickly express repeated multiplication. For instance, the expression \((-2)^{2}\) tells us to multiply -2 by itself, resulting in \(4\). On the other hand, \((-3)^{2}\) requires multiplying -3 by itself, which gives us \(9\).
When evaluating exponents, especially with negative numbers, make sure to apply the exponent to the base in parentheses. The squared sign \(^{2}\) means the number is multiplied by itself once. Understanding this ensures clarity in every step of simplifying numerical expressions.

Substitution and Multiplication

Once you've evaluated the exponents, the next step is substitution and multiplication. This means taking the values you calculated and placing them back into the expression. Our original expression transforms once you substitute the exponents.
For example, substituting gives us: \(3 imes 4 - 2 imes 9\). This new expression requires multiplication first. Calculate each term separately:

• \(3 imes 4 = 12\)
• \(2 imes 9 = 18\)

This ensures our arithmetic sequence is correct. Always handle multiplication before subtraction following the order of operations.

Subtraction of Terms

Subtraction comes after substitution and multiplication in our order of operations. Once you've multiplied, you'll have two results, \(12\) and \(18\). Now is the time to subtract:

• \(12 - 18 = -6\)

This result \(-6\) will be crucial for the next steps. Remember, subtraction could change the sign of your result, as seen here moving from positive to negative.

Evaluating Cubed Numbers

Cubing a number means multiplying it by itself twice more. Here, we're taking \(-6\) and raising it to the power of three \((-6)^{3} \). Start by multiplying \(-6\) by itself:

• \(-6 \times -6 = 36\)

Then, multiply that result by \(-6\) again:

• \(36 \times -6 = -216\)

Thus, cubing \(-6\) gives \(-216\). Always track the negative signs to ensure accuracy. This final step provides the simplified form of the original numerical expression.