Question

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

Short answer

The expression \( x \cdot x \cdot y \cdot y \cdot y \) in exponential form is \( x^{2} \cdot y^{3} \).

Step-by-step solution

Identify the bases and count their repetition

In the given expression, \( x \cdot x \cdot y \cdot y \cdot y \), the bases are x and y. Count how many times x and y occur. There are 2 x's and 3 y's.

Write the expression in exponential form

Exponential notation is written as base^exponent. Write x and y as bases and use the counted repetitions as the exponent. Therefore, x is raised to the power of 2 and y is raised to the power of 3. The expression becomes \( x^{2} \cdot y^{3} \).

Key concepts

Exponential Notation

Exponential notation is an efficient way to express repeated multiplication of the same number. It's composed of two parts: the base and the exponent. The base is the number being multiplied and the exponent tells us how many times to multiply the base by itself. For instance, rather than writing out \( 2 \times 2 \times 2 \times 2 \times 2 \), which can be cumbersome, we write this concisely as \( 2^5 \). The number 2 is the base and the number 5 is the exponent, indicating that 2 should be multiplied by itself a total of five times.

Similarly, in the exercise provided, the variables 'x' and 'y' are multiplied multiple times. Exponential notation simplifies this representation by combining these multiplications into a more compact form. By converting an expression into exponential notation, we can work with it more easily and avoid errors that might come from dealing with lengthy products.

Bases and Exponents

The base in an exponential expression is the value that is being repeatedly multiplied. The exponent, located above and to the right of the base, indicates the number of times the base multiplies itself. The exponent is sometimes referred to as the 'power' or 'index'. In the context of the given exercise, each variable acts as the base in its own exponential expression.

Understanding the role of bases and exponents is crucial when converting standard notation to exponential form. By identifying the bases (in this case, 'x' and 'y') and recognizing that the exponent represents the count of how frequently each base appears, we can easily rewrite expressions with more clarity and less complexity. It is important to note that an exponent of one is often omitted because any number raised to the power of one is the number itself.

Multiplication of Variables

When multiplying variables, if the same variable is multiplied by itself, we can express this operation using exponential form. This is a common step in algebra that simplifies expressions and makes handling variables more straightforward. For example, \( a \times a \) would simply be \( a^2 \).

This concept extends to more complex scenarios where two different variables are being multiplied, as seen in the exercise. When you multiply different variables, you write them side by side with their exponents. As illustrated, \( x \times x \times y \times y \times y \) can be made simpler by reprsenting it as \( x^2 \times y^3 \). This shows that 'x' is multiplied by itself twice, and 'y' is multiplied by itself thrice. The process consolidates the expression into a product of variables, each raised to their respective exponents, making algebraic operations such as expansion, factoring, and simplification much more manageable.