Problem 35 - Algebra 1

Question

Simplify the variable expression. $$-(-4)^{2}(y)$$

Step-by-Step Solution

Answer

The simplified variable expression is \(-16y\).

Step 1: Deal with the exponent

First, consider evaluating \(-(-4)^{2}\). The exponent here 2 tells us to multiply -4 with itself. However, remember that the negative sign outside the bracket is separate from the exponent. So this will result in \(-16\).

Step 2: Multiply the result with y

Then we multiply \(-16\) with \(y\). This gives us \(-16y\).

Key Concepts

Exponents

When dealing with exponents, it's essential to understand that an exponent refers to the number of times a number, known as the base, is multiplied by itself. For example, when you see an expression like \( 4^2 \), that means \( 4 \) is the base, and you are to multiply \( 4 \) by itself twice (\( 4 \times 4 \)), resulting in \( 16 \).

Exponents can also be accompanied by a negative sign, which affects the base differently depending on its placement. If the negative sign is within parentheses, such as \( (-4)^2 \), it means that the negative base is multiplied by itself and the result is positive because a negative times a negative equals a positive (\( -4 \times -4 = 16 \)). However, if the exponent is outside the parentheses, like in \( -4^2 \), then only the number 4 is multiplied by itself, and the negative sign is then applied to the result (\( - (4 \times 4) = -16 \)).

• Remember that the base and its exponent are handled first before applying any other operations.
• If an exponent is negative, it indicates division or a reciprocal, but that is not the case in our current example.

Negative Numbers

Understanding how to work with negative numbers is crucial in algebra. A negative number is a number that is less than zero, represented by a minus sign (\( - \)). When you multiply a negative number by a positive number, the result is negative, and when you multiply two negative numbers, the result is positive.

When simplifying an expression that includes negative numbers, it is important to pay close attention to the signs and the order of operations. In our example, \( -(-4)^2 \), the negative sign in front of the parentheses gets applied after we calculate the exponent. Since \( (-4)^2 = 16 \), adding the negative sign makes it \( -16 \).

• Always perform multiplications or divisions before adding or subtracting.
• The rules of signs are vital: negative times negative equals positive, and negative times positive equals negative.

Algebraic Expressions

An algebraic expression is a mathematical phrase that can contain numbers, operators (like add, subtract, multiply, divide), and variables (like \( x \), \( y \), or \( z \)) but does not have an equality sign. In algebra, we often work to simplify these expressions to understand and solve equations better.

In our exercise \( -(-4)^2(y) \), we are working to simplify the expression step by step. First, we tackle the exponent as explained earlier. Upon obtaining the result of the exponent part, \( -16 \), we then have to multiply this by the variable \( y \). The final simplified form of the algebraic expression is \( -16y \).

• Simplification typically involves combining like terms, handling exponents, and executing arithmetic operations while following the order of operations (PEMDAS/BODMAS).
• Keeping variables intact during simplification is crucial; they represent unknown quantities we might solve for later.