Problem 91 - Elementary and Intermediate Algebra

Question

Evaluate each expression. $$ [6(5)-5(5)]^{3}(-4) $$

Step-by-Step Solution

Answer

The evaluated expression is -500.

Step 1: Solve Inside the Brackets

First, evaluate the expression inside the brackets: \[ 6(5) - 5(5) \]Calculate each multiplication:\[ 6(5) = 30 \]\[ 5(5) = 25 \]Subtract the results:\[ 30 - 25 = 5 \]

Step 2: Raise to the Power of 3

Now, take the result from Step 1, which is 5, and raise it to the power of 3:\[ 5^3 = 5 \times 5 \times 5 = 125 \]

Step 3: Multiply by -4

Finally, multiply the result from Step 2 by -4:\[ 125 \times (-4) = -500 \]

Key Concepts

Exponents

Exponents are a shorthand way to express repeated multiplication. In the given exercise, you encountered an exponent when raising the number 5 to the power of 3. This is denoted as \( 5^3 \). It means 5 multiplied by itself three times.
For better clarity, think of it this way:

• The exponent tells us how many times the base (5 in this case) is used as a factor.
• The base number is like a "building block" that we multiply repeatedly.

Breaking it down, \( 5^3 \) is calculated as:

• First, multiply 5 by 5 to get 25.
• Next, multiply the result by 5 again, which gives you 125.

The exponent rules help us simplify large numbers and make calculations much more manageable. Always remember, the process of dealing with exponents could involve large numbers, so keep the idea of repeated multiplication firmly in mind.

Multiplication

Multiplication in the order of operations (often remembered by the acronym PEMDAS) happens after parentheses but before addition and subtraction. In the original exercise, you calculated two multiplications: \( 6 \times 5 \) and \( 5 \times 5 \).
Let's review the basics of multiplication:

• Multiplication is repeated addition. For example, \( 6 \times 5 \) is the same as adding 5 together six times.
• Start by multiplying the first number (the multiplicand) by the second number (the multiplier).
• In both cases in this exercise, you found the products: \( 6 \times 5 = 30 \) and \( 5 \times 5 = 25 \).

Knowing where multiplication fits in the order of operations is essential to avoid errors in calculations. It ensures that expressions are solved consistently across different problems. Aim to complete any multiplications before moving on to subsequent operations such as subtraction or addition.

Subtraction

Subtraction can seem straightforward, but when combined with other operations, it requires careful attention. In the exercise, after multiplying numbers, subtraction was used to simplify the expression: \( 30 - 25 \). This operation reduced what's inside the brackets before dealing with exponents.
Let's dive deeper:

• Subtraction is taking away a certain amount from another number. Here, it's removing 25 from 30.
• The result, 5, becomes a critical value as it feeds into the exponent component of the problem.
• Subtraction is performed after any calculations inside brackets and any multiplications or divisions in the sequence of operations.

The importance of the order of operations cannot be underestimated. Using subtraction correctly ensures accuracy in simplifying expressions, contributing to the correct execution of more complex math tasks.