Problem 32 - Elementary Algebra

Question

For the following problems, write each of the quantities using exponential notation. $$ (2 y)(2 y) 2 y 2 y $$

Step-by-Step Solution

Answer

Answer: The exponential notation of the given expression is \(16y^4\).

Step 1: Identify the common terms and their exponents

In the given expression \((2y)(2y)2y2y\), we can see that there are 4 copies of the term \(2y\). Each of these terms are being multiplied together.

Step 2: Express the product as an exponent

Since we have 4 copies of the term \(2y\), we can express this as a single term raised to a power of 4: $$ (2y)^4 $$

Step 3: Apply rules of exponents

Since we have each of the terms inside the parenthesis raised to an exponent, we can apply the rules of exponents to find the final form: $$ (2^4)(y^4) $$

Step 4: Calculate the final result

Finally, we compute 2 raised to the power of 4: $$ (16)(y^4) $$ Thus, the given expression can be written in exponential notation as: $$ 16y^4 $$

Key Concepts

Exponent Rules

Understanding exponent rules is essential for simplifying expressions. Exponents, often referred to as powers or indices, are a way of expressing repeated multiplication of a number by itself. For example, in the expression \((2)^4\), the number 2 is the base, and 4 is the exponent. It signifies that 2 should be multiplied by itself 4 times, which calculates to 16. There are several key rules you should remember about exponents:

• Product of Powers Rule: If you multiply the same bases, add their exponents, such as \(a^m \times a^n = a^{m+n}\).
• Power of a Power Rule: If you have an exponent raised to another exponent, multiply the exponents, \((a^m)^n = a^{m \cdot n}\).
• Power of a Product Rule: Distribute the exponent to each factor inside the parentheses, \((ab)^n = a^n \times b^n\).
• Zero Exponent Rule: Any nonzero number raised to the 0 power is 1, like \(a^0 = 1\).

Using these rules can help you simplify complex expressions and solve problems efficiently. By recognizing patterns and applying these rules, the computation becomes much more manageable.

Identifying Common Terms

Identifying common terms in an expression involves recognizing which factors appear repeatedly. This skill is crucial because it allows us to simplify calculations by using exponential notation.In the problem \((2y)(2y)2y2y\), it's clear that the term \(2y\) appears four times in total. When you realize that the same element is repeated, you can use it as a single base raised to the power equal to its occurrences. Recognizing common terms streamlines the simplification process with the power of exponents. This not only makes expressions more concise but also sets the stage for applying more complex operations with ease.

Multiplication of Variables

Multiplication of variables follows similar rules to those of numbers. When variables are multiplied together, their exponents can be combined if the bases are the same.In an expression, such as \((2y)^4\), both the number 2 and the variable \(y\) are separately being multiplied. Here’s how you could break it down:

• First, recognize \((2y)^4\) as \((2^4)\cdot(y^4)\). This means each factor inside the parenthesis is raised to the power of 4.
• Compute \(2^4\) which results in 16.
• The \(y\) has its own exponent which is 4, showing that \(y\) is multiplied by itself four times (\(y \cdot y \cdot y \cdot y\)). This simplifies to \(y^4\).

Utilizing multiplication of variables with exponents permits simplification, not only making the expression neater but also enabling more straightforward computations later in mathematical processes.