Problem 56 - Algebra 1: Concepts and Skills

Question

Write the expression as a single power of the base. (Lesson 8.1) $$y^{3} \cdot y$$

Step-by-Step Solution

Answer

The expression \(y^{3} \cdot y\) can be written as a single power of the base 'y' as \(y^{4}\).

Step 1: Identifying the Base and Exponents

In the given expression \(y^{3} \cdot y\), the base 'y' is repeated twice, once with an exponent of 3 and once with an implied exponent of 1.

Step 2: Applying the Exponent Addition Rule

According to the rule \(a^{m} \cdot a^{n} = a^{m+n}\), the same bases (in this case, the 'y's) are multiplied by adding their exponents together. Therefore, the expression becomes \(y^{3+1}\).

Step 3: Simplifying the Exponent

Adding the exponents gives \(y^{4}\).

Key Concepts

Base and Exponent Identification

When dealing with expressions that involve exponents, the first step is to identify both the base and the exponent in each part of the expression. In the exercise provided, the expression is \(y^{3} \cdot y\). Here, the "base" is the repeated quantity or symbol, which is 'y'. Notice that the base remains the same throughout the expression.

Exponents indicate how many times the base is multiplied by itself. For \(y^{3}\), the '3' is the exponent, showing that the base 'y' is used as a factor three times: \(y \cdot y \cdot y\). In the term 'y' without any written exponent, there is an implied exponent of 1, meaning \(y^{1} = y\). Understanding this setup is crucial as the foundation for further manipulation of the expression.

Multiplying Powers with the Same Base

A key aspect of working with powers is knowing how to multiply them when they have the same base. The rule to remember is: when you multiply powers that share the same base, you keep the base and add the exponents. This is known as the Exponent Addition Rule and can be written as:
\[a^{m} \cdot a^{n} = a^{m+n}\]
In the exercise, both terms have the same base 'y'. So, by applying the rule, you simply add the exponents: \(3\) from \(y^{3}\) and the implied \(1\) from \(y^{1}\). Therefore, you get:
\[y^{3} \cdot y^{1} = y^{3+1}\]
This means you're effectively counting how many times "y" appears in total as a factor in the combined expression.

Simplifying Exponents

Once you have used the Exponent Addition Rule, simplifying the expression becomes the final straightforward step. After identifying and adding up the exponents in \(y^{3+1}\), perform the addition:
\[3 + 1 = 4\]
So, the expression can be simplified to \(y^{4}\). This result tells us that the base 'y' is multiplied by itself four times.

Remember: Simplifying exponents is about combining terms in the most reduced form while still maintaining mathematical accuracy. This step showcases the beauty of exponent rules, reducing potentially complex expressions into elegant simplicity. By doing so, you make calculations and further manipulations easier to handle.