Problem 57 - Algebra 1

Question

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

Step-by-Step Solution

Answer

The simplified form of the variable expression is \(16t^2\).

Step 1: Simplify the Exponentiation

First, always remember that the exponentiation should be performed first in the operation. Here, the exponentiation is \((-4)^2\). When a negative number is raised to an even power, the result is always positive. Hence, \((-4)^2 = 16\).

Step 2: Multiply by the Variables

Second, take the result from step 1, which is 16, and multiply it by \(t*t\) which is \(t^2\), meaning t multiplied by t. Thus, it will become \(16*t^2\).

Key Concepts

Exponentiation

Understanding exponentiation is crucial when working with mathematical expressions. It refers to the process of raising a number, known as the base, to the power of an exponent. The exponent, located as a superscript to the right of the base, tells us how many times the base is to be multiplied by itself. For example, when you see an expression like \(3^4\), it means you should multiply 3 by itself 4 times, which is \(3 \times 3 \times 3 \times 3 = 81\).

In the context of our exercise, \( (-4)^2 \), the base is -4 and the exponent is 2. This instructs us to multiply -4 by itself one time (since the initial -4 is considered the first instance), resulting in \( (-4) \times (-4) = 16 \). Remember that exponentiation takes precedence over other operations like multiplication or addition, so it should always be performed first.

Variables Multiplication

Multiplying variables follows a similar pattern to multiplying numbers, but with a focus on combining the variables' exponents when they are the same. When two expressions with the same variable base are multiplied, you add the exponents to find the new power of that base. For instance, \( x \times x \) or \( x^1 \times x^1 \) becomes \( x^{1+1} = x^2 \).

In our exercise, after simplifying the exponentiation, we multiply the result by \( t \times t \) or \( t^1 \times t^1 \), which simplifies to \( t^{1+1} = t^2 \). So, multiplying \(16 \), which is a constant, by \( t^2 \), gives us \( 16t^2 \) as the final simplified variable expression. Always ensure the constants and variables are correctly accounted for to prevent any mistakes.

Negative Numbers to Even Powers

When dealing with negative numbers as bases in exponentiation, the power or exponent plays a significant role in determining the sign of the result. A negative number raised to an even exponent yields a positive result because multiplying two negative numbers results in a positive product. For example, \( (-2)^2 = (-2) \times (-2) = 4 \).

This property is evident in our original exercise with \( (-4)^2 \). Since 2 is an even number, the squared negative base becomes a positive number. Remember that this only holds true for even exponents. Negative bases raised to odd exponents will result in a negative value, such as in \( (-3)^3 = -27 \). Keeping this rule in mind is essential when simplifying variable expressions with negative bases and exponents.