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Chapter 4: Problem 4

Write each expression using exponents. $$9 \cdot x \cdot x \cdot y \cdot y \cdot y \cdot y$$

Short Answer

Expert verified
The expression \(9 \cdot x \cdot x \cdot y \cdot y \cdot y \cdot y\) in exponent form is \(9^1 \cdot x^2 \cdot y^4\).

Step by step solution

01

Identify repeated factors

Expression contains factors which are multiplied together. By closely looking, 9 is multiplied once, x is multiplied twice, and y is multiplied four times. So, identify 9, x and y as bases and the count of their multiplication as their respective exponents.
02

Write expression using exponents

Bases will remain the same, however, replace the repeated multiplication with exponentiation - Write the base and annex the total number of repeated multiplication as the exponent. This gives the expression as \(9^1 \cdot x^2 \cdot y^4\).
03

Simplify expression if needed

Usually, if the exponent is 1, it's not written explicitly. However, the problem does not state whether to simplify or not, so we鈥檒l keep it. Therefore, the final expression in exponential form is \(9^1 \cdot x^2 \cdot y^4\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Algebraic Expressions
Algebraic expressions are mathematical phrases that can include numbers, variables, and operations. In the exercise, the expression is made up of numbers and variables which are multiplied together: \(9\), \(x\), and \(y\). An algebraic expression does not have an equal sign, unlike an equation. This means it represents a value depending on the variable values assigned.
Variables such as \(x\) and \(y\) can take various values. This versatility makes algebraic expressions useful for generalizing mathematical relationships and patterns.
The Role of Multiplication in Expressions
Multiplication is one of the basic operations we use in algebra. In our exercise, multiplication is used to combine the numbers and variables. The expression \(9 \cdot x \cdot x \cdot y \cdot y \cdot y \cdot y\) shows how each part interacts.
Instead of adding like terms together, multiplication is repeated addition. It combines the quantities in a compact form. You will often see multiplication in expressions because it helps create more manageable and meaningful forms of the original values.
The Power of Exponentiation
Exponentiation is a way to express repeated multiplication of the same number or variable. The expression \(x \cdot x\) is simplified to \(x^2\) because \(x\) is multiplied by itself twice. Similarly, \(y \cdot y \cdot y \cdot y\) becomes \(y^4\) as \(y\) is repeated four times.
Exponentiation helps to make expressions more concise. It also simplifies calculations, especially with larger numbers. By using exponents, we can quickly calculate the results of repeated multiplication without writing each factor.
Simplification of Expressions
Simplification involves rewriting an expression in a way that is easier to understand or work with. In our exercise, we use exponents to make the expression \(9 \cdot x \cdot x \cdot y \cdot y \cdot y \cdot y\) into \(9^1 \cdot x^2 \cdot y^4\).
Sometimes, simplification means removing unnecessary parts, like when the exponent is 1. However, in this case, we're keeping \(9^1\) for clarity, although it can be written simply as \(9\). Simplifying expressions is often crucial for solving more complex problems, as it leads to clearer, more usable forms.

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