Problem 80 - Algebra 1

Question

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

Step-by-Step Solution

Answer

The simplified form of \(-(-3)^{2}(-y)\) is \(9y\).

Step 1: Simplify the Expression Inside Parentheses

Start by simplifying \((-3)^{2}\). The exponent tells us to multiply -3 by itself. The result is \((-3)\times (-3) = 9\). The expression becomes: \(-9(-y)\).

Step 2: Apply Distributive Property

Next, apply the negative sign referred as distributive property to the remaining part of the expression. The result is \(-9 \times -y = 9y\). This is because multiplying two negative values yield a positive value.

Step 3: Write Final Simplified Expression

The expression \(-(-3)^{2}(-y)\) has now been simplified to \(9y\).

Key Concepts

Exponents

Exponents are a mathematical operation that involves raising a number to a specified power. This means you multiply the base number by itself a number of times equal to the exponent. For example, the expression \((-3)^2\) means that you multiply \(-3\) by itself, giving \((-3) \times (-3)\).

• The base is the number being multiplied.
• The exponent tells you how many times to multiply the base by itself.

In our example, \((-3)^2 = 9\) because multiplying two negative numbers results in a positive product. Always pay attention to parentheses, as they indicate that the negative sign is part of the base. Without parentheses, \(-3^2\) would be interpreted as \(-(3^2)\) which equals \(-9\).So, remember that parentheses can change the entire value of your calculation.

Distributive Property

The distributive property is a fundamental concept in algebra. It simplifies expressions when multiplying a single term by a sum or an expression inside parentheses. It looks like this: \(a(b + c) = ab + ac\).The idea is to distribute the multiplication over each term inside the parentheses.

• To distribute, multiply the term outside the parentheses with each term inside it.
• Use this property to simplify algebraic expressions efficiently.

In our exercise, when you have \(-(-3)^2(-y)\), you first simplify inside the parentheses and then distribute. After finding \(-9\) from \((-3)^2\), distribute the negative sign, leading to \(-9\times -y = 9y\). Understand this property can greatly reduce the complexity and steps needed while solving similar expressions.

Multiplication Rules

Multiplication rules allow you to apply logical steps when working with numbers and expressions. These rules help simplify calculations, especially when signs are involved. Here are essential points to remember:

• Multiplying two positive numbers gives a positive result (e.g., \(2 \times 3 = 6\)).
• Multiplying two negative numbers results in a positive product (e.g., \((-2) \times (-3) = 6\)).
• Multiplying a positive number by a negative number results in a negative product (e.g., \(2 \times (-3) = -6\)).

In the exercise, when distributing \(-9\) over \(-y,\) the result is positive, yielding \(9y\). Being consistent with these basic rules is crucial while solving algebraic expressions with multiple layers and negative signs.

Negative Numbers

Understanding negative numbers is vital in solving expressions involving subtraction, debt, or inversion concepts. A negative number indicates a value below zero, and applying it requires careful handling of signs.

• Two negative numbers multiplied together yield a positive result, as negatives cancel out each other.
• When adding a positive and negative number, you subtract their absolute values, retaining the sign of the larger absolute value number.
• Subtraction of a negative number is equivalent to adding a positive number (e.g., \(5 - (-3) = 5 + 3 = 8\)).

In the given expression, the initial negative outside \(-(3^2)\) resulted in converting the entire product to \(-9,\) but going forward, multiplying by another negative, \(-y\), converts it back to positive, thus giving \(9y\). Handling negative signs correctly influences the outcome and interpretation of mathematical expressions.