Problem 23 - Elementary Algebra

Question

For the following problems, write each of the quantities using exponential notation. \(2 \cdot 3 \cdot 3 \cdot 3 \cdot 3 x x y y y y y\)

Step-by-Step Solution

Answer

Question: Rewrite the expression \(2 \cdot 3 \cdot 3 \cdot 3 \cdot 3 \cdot x \cdot x \cdot y \cdot y \cdot y \cdot y \cdot y\) using exponential notation. Answer: \(2\cdot 3^4 \cdot x^2 \cdot y^5\)

Step 1: Determine the quantities that are repeated

The given expression is \(2 \cdot 3 \cdot 3 \cdot 3 \cdot 3 \cdot x \cdot x \cdot y \cdot y \cdot y \cdot y \cdot y\). Observe the repeated quantities: we have four \(3\)s, two \(x\)s, and five \(y\)s.

Step 2: Write each repeated quantity using exponents

Since we have four \(3\)s, we can write it as \(3^4\). There are two \(x\)s, so it becomes \(x^2\). Finally, we have five \(y\)s, which can be written as \(y^5\).

Step 3: Combine all quantities

Now that we have rewritten each repeatable quantity using exponents, we will put them together. Multiplying the original \(2\) with the newly calculated exponential quantities, our final expression becomes: \(2\cdot 3^4 \cdot x^2 \cdot y^5\).

Key Concepts

Understanding Algebraic Expressions

Algebraic expressions are the backbone of algebra and comprise numbers, variables, and arithmetic operations (addition, subtraction, multiplication, and division). These expressions can represent real-world scenarios, such as the total cost of apples if each costs \(2 and you buy 'x' amount, which can be written as \)2x.

Algebraic expressions become especially powerful when they include exponents to indicate repeated multiplication. For example, the expression for the volume of a cube with sides of length 'a' is shown as \(a^3\), compactly expressing \(a \times a \times a\).

Identifying and simplifying such expressions help in problem-solving and are fundamental in more advanced mathematical concepts.

Exponents and Their Rules

Exponents, also known as powers, are a shorthand notation for expressing repeated multiplication of the same number or variable. For instance, \(5^3\) (read as 'five to the third power' or 'five cubed') means \(5 \times 5 \times 5\). It's important to note that \(5^1 = 5\) and any number to the power of 0 is 1, denoted by \(5^0 = 1\).

Some key rules of exponents include:

• The product rule: \(a^m \times a^n = a^{m+n}\)
• The power rule: \((a^m)^n = a^{m \times n}\)
• The quotient rule: \(a^m / a^n = a^{m-n}\) when \(m > n\) and 1 when \(m = n\)

Understanding and correctly applying these rules are crucial in manipulating algebraic expressions and solving complex equations.

Multiplication of Variables

Multiplication of variables follows the same fundamental principles as multiplying numbers, but with a focus on combining 'like terms'. Like terms are variables that have the same base and exponent. For example, multiplying \(x^2\) by \(x^3\), since they have the same base 'x', uses the product rule: \(x^2 \times x^3 = x^{2+3} = x^5\).

When terms are not like, such as different variables or different exponents, they're multiplied as they are: \(2x^3 \times 3y^2 = 6x^3y^2\). Simplifying expressions by combining like terms or expanding products is a pivotal skill in algebra that leads to solving equations and understanding polynomial functions.