Problem 52
Question
For the following exercises, simplify the expression. \(5^{2}-4(3 x)\)
Step-by-Step Solution
Verified Answer
The simplified expression is \(25 - 12x\).
1Step 1: Calculate the Exponentiation
First, calculate the exponentiation in the expression. Here, we have the term \(5^2\). Compute \(5^2\) which results in \(25\). So, the expression becomes \(25 - 4(3x)\).
2Step 2: Simplify the Multiplication Inside the Parentheses
Next, identify and simplify the multiplication inside the parentheses. The expression \(4(3x)\) needs to be simplified. Multiply \(4\) by \(3x\), which gives \(12x\). Substitute this back into the expression to get \(25 - 12x\).
3Step 3: Simplify the Final Expression
After performing the multiplication and removing the parentheses, simplify the expression if possible. Here, the expression \(25 - 12x\) is already simplified as there are no like terms to combine.
Key Concepts
ExponentiationSimplificationParentheses in Algebra
Exponentiation
Exponentiation is a way to simplify multiplication when the same number is multiplied by itself multiple times. In our example, this is seen with the term \(5^2\). This means we multiply 5 by itself, which is \(5 \times 5\). Calculate this to get 25.
Understanding exponentiation is important because it often serves as the first step in simplifying algebraic expressions.
When you see an exponent, you should always resolve it before moving on to other parts of the expression. This follows the established order of operations in mathematics.
Understanding exponentiation is important because it often serves as the first step in simplifying algebraic expressions.
When you see an exponent, you should always resolve it before moving on to other parts of the expression. This follows the established order of operations in mathematics.
Simplification
Simplification involves reducing an expression to its simplest form. This makes it easier to understand and solve. In the example, after calculating the exponentiation, you're left with the expression \(25 - 4(3x)\).
Simplification here requires looking into the expression inside the parentheses, which is \(4(3x)\). Perform multiplication first to combine terms efficiently. \(4 \times 3x\) results in \(12x\). Substitute this to update the expression to \(25 - 12x\).
Once you've done these operations, the expression is in its simplest form as \(25 - 12x\), meaning no further simplification is possible since there are no like terms to combine.
Simplification here requires looking into the expression inside the parentheses, which is \(4(3x)\). Perform multiplication first to combine terms efficiently. \(4 \times 3x\) results in \(12x\). Substitute this to update the expression to \(25 - 12x\).
Once you've done these operations, the expression is in its simplest form as \(25 - 12x\), meaning no further simplification is possible since there are no like terms to combine.
Parentheses in Algebra
Parentheses in algebra dictate the order in which calculations should be performed. They show which operations should be done first, ensuring that the expression is resolved correctly.
In our sample expression, parentheses are used around \(3x\) indicating the multiplication must happen with the 4 outside. This is crucial to preserving the intended operations in the expression.
In our sample expression, parentheses are used around \(3x\) indicating the multiplication must happen with the 4 outside. This is crucial to preserving the intended operations in the expression.
- Perform all operations inside the parentheses first.
- Once the calculations within the parentheses are complete, remove them and continue solving.
Other exercises in this chapter
Problem 52
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