Problem 18 - Elementary Algebra

Question

For the following problems, write each of the quantities using exponential notation. 3 squared times \(y\) to the fifth

Step-by-Step Solution

Answer

Question: Rewrite the expression "3 squared times y to the fifth" using exponential notation. Answer: (3^2)(y^5)

Step 1: Rewrite 3 squared in exponential notation

Since "3 squared" means 3 raised to the power of 2, we can write it as 3^2.

Step 2: Rewrite \(y\) to the fifth in exponential notation

Since "\(y\) to the fifth" means \(y\) raised to the power of 5, we can write it as y^5.

Step 3: Multiply 3 squared and \(y\) to the fifth

As both expressions are already in exponential notation, we can simply multiply them together: (3^2)(y^5).

Step 4: Final Answer

The final expression in exponential notation is (3^2)(y^5).

Key Concepts

Exponents

Exponents are a way to express repeated multiplication of the same number by itself. They signify how many times a number, known as the "base," is multiplied by itself. For example, in the expression \(3^2\), the number 3 is the base, and it is multiplied by itself 2 times: \(3 \times 3\). The number of times the base is multiplied is indicated by the exponent, which is the small number written to the right and slightly above the base.
In general terms, an exponent is written as \(a^n\), where "a" is the base and "n" is the exponent. This denotes that "a" is used as a factor "n" times. Understanding exponents is crucial for simplifying and manipulating algebraic expressions, as they often appear in various mathematical contexts.
When dealing with different bases and exponents, it's important to note that each base-exponent pair stands on its own and cannot be directly combined with others unless certain mathematical rules are used.

Algebraic Expressions

Algebraic expressions are collections of numbers, variables, and mathematical operations used to represent a specific value or set of values. These include constants, variables (often represented by letters like \(x\) or \(y\)), and the arithmetic operations of addition, subtraction, multiplication, and sometimes division.
A simple example of an algebraic expression is \(3x + 5\). Here, 3 is the coefficient of the variable \(x\), and 5 is a constant. The arithmetic operation involves adding a product of a number and a variable to another number.
It is essential to understand that algebraic expressions allow for a concise representation of mathematical ideas and patterns. They enable us to express relationships and changes among quantities quickly. Expressions can include other mathematical concepts such as exponents, which indicate repeated multiplication.
Algebraic expressions become simpler to deal with when you recognize patterns and rules applicable to their structures, such as distributive laws, factoring, and combining like terms.

Multiplication of Powers

Multiplication of powers is a mathematical operation that combines numbers or variables with exponents. When multiplying powers with the same base, you can add their exponents instead of multiplying the bases directly. For instance, \(a^m \times a^n = a^{m+n}\) shows that if the base "a" is the same, you just add the exponents, "m" and "n".
This is particularly useful for simplifying expressions that involve multiple sets of powers. In cases where the bases are different, such as \(3^2\) and \(y^5\), you simply keep them side by side in the expression, like \((3^2)(y^5)\). There is no direct simplification for different bases, so the expression already represents the multiplication fully.
Understanding multiplication of powers is a foundational skill in algebra, as it streamlines working with exponential expressions and is crucial in solving higher-level algebraic equations. Always be sure to check the bases of the powers involved to apply the correct operations.