Problem 51 - Algebra 1: Concepts and Skills

Question

Evaluate the expression for the given value of the variable. \((5 y)^{5}\) when \(y=2\)

Step-by-Step Solution

Answer

The evaluated expression equals 100,000.

Step 1: Substitution

Substitute the given value of \(y = 2\) into the expression. This gives us \((5*2)^5\).

Step 2: Solve Multiplication Inside Parentheses

By rules of precedence, also known as BODMAS or PEMDAS, multiplication comes before exponentiation. So, first we multiply 5 by 2 which equals 10. Bringing down the rest of the expression, this gives us \(10^5\).

Step 3: Exponentiation

Compute \(10^5\), which equals 100,000.

Key Concepts

Understanding Substitution in Algebra

Substitution is like a mathematical version of a costume change. In algebra, it means replacing variables with their numerical values to simplify expressions. In our exercise, for instance, we have the variable 'y' in the expression (5y)^5, and we're told that y is actually a 2 in disguise. So, we swap y for 2, getting (5*2)^5. It's like telling 'y' to step aside while we bring in the number 2 to take its place.

Substitution is fundamental because it allows us to evaluate algebraic expressions and understand how different values affect the outcome. Imagine you're baking and the recipe calls for 'sugar', but you need to know exactly how much. That's why we substitute: to see the quantity and work with it directly. Remember that correct substitution makes the rest of the calculations much smoother and more accurate.

Order of Operations: The Math Symphony

Following the order of operations in math is crucial, just like following a recipe step by step is for a successful dish. This 'recipe' is often remembered by the acronym PEMDAS or BODMAS, representing Parentheses/Brackets, Exponents/Orders, Multiplication/Division, and Addition/Subtraction.

In our practice problem, once we substitute 2 for 'y', we get (5*2)^5. Now, attention to the order is paramount. We must multiply the numbers inside the parentheses before we tackle the exponent - that's our multiplication before exponentiation - resulting in 10^5. Think of it like tuning the instruments (multiplication) before the orchestra plays its symphony (exponentiation). Nailing this sequence ensures harmonious results every time.

Exponentiation: Multiplication on Repeat

Exponentiation is when we raise a number to the power of another. It's a fancy way of saying we're multiplying a number by itself several times. In our example, 10^5 means we're having a multiplication party with 10 as the guest of honor; it's multiplied by itself exactly four more times. The result? A whopping 100,000. Exponentiation expresses growth and expansion concisely, like saying ‘I told you a secret, and you told five others’; it amplifies quickly!

To excel in exponentiation, remember that it can escalate numbers rapidly, and thus, it has a significant impact on your calculation's outcome. Always double-check your exponent work to avoid a small mistake ballooning into a big one.