Problem 54 - Algebra 1

Question

Simplify the variable expression. $$(-2)^{3}(b)$$

Step-by-Step Solution

Answer

The simplified expression is -8\(b\).

Step 1: Interpret the Expression

The given expression is \((-2)^{3}(b)\). This expression contains the cube of -2 multiplied by the variable \(b\). The exponent applies only to -2, not to \(b\).

Step 2: Apply Exponent rules

When any negative number is raised to an odd exponent, the result is also a negative number. Hence, \((-2)^3\) will be -8. So, now the expression becomes -8 times \(b\) or -8\(b\).

Step 3: Simplifying the expression

As the expression is now -8\(b\), there are no other simplifications to be done. The expression is already simplified.

Key Concepts

Exponents

Exponents are a way of expressing repeated multiplication of a number by itself. For example, in the expression \((-2)^3\), the exponent is 3, which means we multiply -2 by itself three times: \((-2) \times (-2) \times (-2) = -8\). Understanding this concept is crucial because the sign of the base (negative in this case) affects the final result. Exponents can be:

• Positive: indicating how many times to multiply the base by itself
• Even or odd: even exponents yield positive results if the base is negative, while odd exponents yield negative results.

In our problem, the base (-2) is negative and raised to an odd exponent (3), therefore, the product is negative. It's essential to remember that the exponent only affects the base directly attached to it, and not any other part of the expression, like a variable term.

Negative Numbers

Negative numbers are less than zero on the number line, and they can be a bit tricky when it comes to multiplication and division. For negative numbers:

• When multiplied by a positive number, the result is negative.
• When multiplied by another negative number, the result is positive, as two negatives make a positive.
• Raised to an odd exponent, the result is negative because the multiplication is repeated an odd number of times.

In our expression, (-2)^3, we're multiplying a negative number by itself three times. This results in a negative outcome, (-8), due to the odd exponent. It's vital to handle negative numbers carefully in expressions and equations, especially when dealing with exponents.

Variable Expressions

Variable expressions include variables combined with numbers and arithmetic operations. In the expression (-8) b, "b" is a variable, meaning it can represent different values. Variable expressions allow for flexibility in algebra, as they can represent a wide range of numeric values depending on the variables involved.

• The coefficient, here (-8), shows how many times the variable is multiplied.
• Simplifying variable expressions involves combining like terms and applying arithmetic rules while respecting the operations performed on numbers.

In our case, once (-2)^3 was simplified to (-8), the remaining expression was simply -8b, indicating (-8) multiplied by whatever value "b" takes on. It's important to fully simplify variable expressions to make calculations easier and more understandable.