Question

Write the expression in exponential form. $$5 y \cdot 5 y \cdot 5 y \cdot 5 y$$

Short answer

The expression \(5y \cdot 5y \cdot 5y \cdot 5y\) in exponential form is \((5y)^4\).

Step-by-step solution

Identify the base

The base of the expression \(5y\) is being multiplied four times, so we identify \(5y\) as the base.

Count the number of times the base is being multiplied

We see that the base \(5y\) is being multiplied by itself four times, so the exponent is 4.

Write the expression in exponential form

Since we identified \(5y\) as the base and 4 as the exponent, we can write the expression as \((5y)^4\).

Key concepts

Base and Exponent

Understanding the base and exponent is a fundamental concept in algebra, especially when dealing with exponential form. In an expression like \(5y \cdot 5y \cdot 5y \cdot 5y\), the term that is multiplied by itself, which is \(5y\), is called the base. The number of times the base is used as a factor determines the exponent. In this case, since \(5y\) is multiplied four times, the exponent is 4. The exponent is written in superscript to the right of the base.

Here's a simple representation: \(base^{exponent}\). Recognizing the base and exponent helps in simplifying expressions and understanding growth patterns in functions. It's also essential for tackling more complex problems such as those involving exponential growth or decay.

Writing Expressions in Exponential Form

When you encounter repeated multiplication of the same term, writing expressions in exponential form can simplify the notation and make calculations easier. Chances are, if you've ever dealt with large numbers or patterns of growth, you've come across an expression that can be written in exponential form. For example, in the expression \(5y \cdot 5y \cdot 5y \cdot 5y\), we can condense it by using an exponent, based on how many times the term \(5y\) appears. The process involves identifying the base—here, it's \(5y\), since that's the term being repeated—and counting how many times it's multiplied by itself, which gives us the exponent. Instead of writing the long repetitive multiplication, we use the notation \( (5y)^4 \), which tells us \(5y\) is multiplied by itself four times. This method is not only neater, it's also crucial when you begin working with more abstract algebra and functions.

Multiplication of Like Terms

Multiplying like terms, terms that have the same variable raised to the same power, is a key skill in simplifying algebraic expressions. If we take our example \(5y \cdot 5y \cdot 5y \cdot 5y\), the repeated term \(5y\) is known as a like term because the variable \(y\) doesn't change its exponent across the multiplications—it always has the same exponent, which is implicitly 1 (since any number to the power of 1 is itself).

When you're multiplying like terms, you multiply the coefficients (in this case, the number 5) and keep the variable part the same. The exponential form comes in handy here because it encapsulates the multiplication of coefficients. It's crucial to get comfortable with this process as it paves the way for understanding polynomials, exponential functions, and even simplifying equations in calculus.