Problem 25 - Algebra 1: Concepts and Skills

Question

Evaluate the expression for the given value of the variable. ( 4)\((n)\) when \(n=5\)

Step-by-Step Solution

Answer

So, the evaluated expression \( (2 x)^{3}\) when \(x=5\) is 1000.

Step 1: Substitute the Given Value

Replace \(x\) in the expression \((2 x)^{3}\) with the value given, which is 5. Therefore, the expression becomes \((2 \cdot 5)^{3}\).

Step 2: Simplify Inside the Parentheses

Calculate the multiplication inside the parentheses, giving \((10)^{3}\).

Step 3: Apply Exponent Rules

Raise the value inside the parentheses, 10, to the power of 3 (\(10 \cdot 10 \cdot 10\)), which results in 1000.

Key Concepts

Substitution in Algebraic Expressions

Understanding how to work with algebra often begins with mastering the concept of substitution. Substitution is like a game of matching, where we replace variables with specific numerical values. Take the expression \( (2x)^3 \), for instance. When we're told that \( x = 5 \) our next move is to swap every \( x \) with \( 5 \) as if exchanging puzzle pieces.

This direct replacement is the first critical step: Step 1: Replace \( x \) with \( 5 \) to get \( (2 \cdot 5)^3 \). Why do we do this? Because we are preparing the expression so that we can evaluate it—a pivotal stage in algebra where expressions become concrete numbers. Simple, right?

But remember, in the game of algebra, careful attention is necessary. Stick to one substitution at a time to avoid mistakes, and always follow the order of operations even in the substitution stage. This will keep the calculation correct and lead you smoothly to the next step of simplifying.

Simplifying Algebraic Expressions

Simplifying is all about making things easier to understand. Our brains appreciate simplicity, and algebraic simplification allows us to turn complex expressions into simple, solvable ones. To simplify an expression, we perform all possible arithmetic operations.

So, let’s roll up our sleeves for Step 2: Once you have replaced \( x \) with \( 5 \) to get \( (2 \cdot 5)^3 \) you perform the multiplication inside the parentheses first—it's all about following the order of operations, or PEMDAS if you prefer catchy acronyms. After we multiply \( 2 \cdot 5 \), we get \( 10 \) and the expression now looks like \( 10^3 \). See how it’s getting tidier?

Picturing simplification as tidying up, where we clear away unnecessary steps to get to the simplest form of an expression, might help. Simplifying isn’t just a part of solving; it’s also about clean expression, which is both aesthetically pleasing and excellent for understanding the underlying mathematics.

Exponent Rules

Once you’ve simplified your algebraic expression, it’s time to tackle the exponents. Exponents can be tricky, but remembering the basic rules makes them much less daunting. The key is to understand that an exponent tells us how many times to multiply a number by itself.

Here we are, at Step 3: We raise \( 10 \) to the power of 3, which means we multiply \( 10 \times 10 \times 10 \), and voila—1000 is our answer. This is simply the third power of 10. Understanding exponent rules takes practice, but when you see an expression like \( 10^3 \), all you need to remember is to multiply the base, \( 10 \) in this case, by itself however many times the exponent indicates.

The tricks don’t stop there, though—there are rules about multiplying exponents, dividing them, and even dealing with negative exponents. Looking for patterns in the numbers can help. For instance, \( 10^3 = 1000 \) and that has three zeros, the same as the exponent three. Spotting patterns like this reinforces your understanding and makes working with exponents less about memorization and more about grasping the concept.