Problem 39 - College Algebra

Question

For the following exercises, simplify the expression. $$ 2 y-(4)^{2} y-11 $$

Step-by-Step Solution

Answer

The simplified expression is \( -14y - 11 \).

Step 1: Calculate the Power

First, calculate the value of the exponent in the expression. Here, we have \( (4)^2 \). Calculate this:\[ (4)^2 = 16 \]

Step 2: Substitute the Power

Now, substitute the calculated value back into the expression. The original expression is:\( 2y - (4)^2y - 11 \).After substitution, it becomes:\( 2y - 16y - 11 \).

Step 3: Combine Like Terms

Next, combine the like terms. Both \( 2y \) and \( -16y \) are like terms:\[ (2y - 16y) = -14y \].So, the expression simplifies to:\( -14y - 11 \).

Step 4: Final Simplified Expression

Finally, confirm that the expression is fully simplified. There are no further like terms or operations to perform, so the simplified expression is:\( -14y - 11 \).

Key Concepts

Exponents

Exponents play a crucial role in algebraic simplification. They represent repeated multiplication of a number by itself. For instance, an exponent denotes how many times a base number is multiplied by itself.
When dealing with exponents, such as in our exercise, we have to first compute the power before combining terms. In \[ (4)^2 \], the base 4 is raised to the power of 2, meaning 4 multiplied by itself. Calculating gives us 16. Simplifying exponents first is important because it transforms the expression into a simpler form that is easier to work with later.

• Identify the base number and the exponent.
• Multiply the base by itself as many times as indicated by the exponent.
• Replace the power in the original expression with the new computed value.

By grasping the operation of exponents, you ensure a smooth simplification process because it reduces complex expressions into understandable parts.

Like Terms

Like terms are terms within an algebraic expression that have the same variable raised to the same power. Recognizing and combining like terms is a fundamental skill in expression simplification.
In the given expression, terms like \( 2y \) and \(-16y \) are considered like terms because they both include the variable y raised to the first power. When you simplify such expressions, focus on rearranging and combining these terms.

• Identify terms with the same variable and exponent.
• Combine their coefficients (the numbers in front of the variables).
• Simplify to make the expression as compact as possible.

In our exercise, combining \( 2y \) and \(-16y \) reduces to \(-14y \). This reduction makes the expression simpler and easier to interpret or solve.

Expression Simplification

Expression simplification involves reducing a complex algebraic expression to its simplest form. This process combines understanding exponents, like terms, and basic operations like addition and subtraction.
Every simplification process starts by addressing each separate element, such as calculating exponents first and then combining like terms. This structured approach ensures nothing is overlooked. A simplified expression should not contain any further operable terms or reducible factors.

• Calculate any exponential terms first to transform the expression.
• Identify and combine like terms to reduce the complexity.
• Ensure no further simplification is possible (no like terms or unresolved operations remain).

In our problem, the original expression \( 2y - (4)^2y - 11 \) first simplifies the exponential term, then combines like terms into \(-14y - 11\). With all operations complete, the expression reaches its final simplified form, making it easy to understand and work with in further calculations.