Problem 62 - Elementary Algebra

Question

What is the GCF of \(y^{4}, y^{5}\), and \(y^{10} ?\) Write a general rule that tells you how to find the GCF of \(y^{a}, y^{b},\) and \(y^{c}\).

Step-by-Step Solution

Answer

The GCF is \(y^{4}\). To find the GCF of \(y^{a}\), \(y^{b}\), and \(y^{c}\), use the smallest exponent among \(a, b, c\).

Step 1 - Identify the exponents

Note the exponents of the given terms: For \(y^{4}\), the exponent is 4. For \(y^{5}\), the exponent is 5. For \(y^{10}\), the exponent is 10.

Step 2 - Compare the exponents

To determine the Greatest Common Factor (GCF), compare the exponents. The GCF will be determined by the smallest exponent since all terms share the base \(y\).

Step 3 - Identify the smallest exponent

Here, the exponents are 4, 5, and 10. The smallest of these exponents is 4.

Step 4 - Write the GCF

Since the smallest exponent is 4, the GCF of \(y^{4}\), \(y^{5}\), and \(y^{10}\) is \(y^{4}\).

Step 5 - General Rule for GCF of powers

To find the GCF of \(y^{a}\), \(y^{b}\), and \(y^{c}\), identify the smallest exponent among \(a\), \(b\), and \(c\). The GCF will be \(y^{\text{min}(a, b, c)}\).

Key Concepts

Greatest Common Factor

The Greatest Common Factor (GCF) is the largest factor that divides two or more numbers or terms. When you're working with algebraic expressions like the ones in our exercise, understanding the GCF helps in simplifying expressions and solving equations.

For example, given the terms with exponents such as \(y^4, y^5,\text{ and } y^{10}\), you need to find the highest power of each shared factor that can divide each term. This tells you how many times a given variable (in this case, \(y\)) can be factored out from all the terms.

Exponents

Exponents refer to the power to which a number or variable is raised. They tell you how many times to multiply the base by itself. For example, in the term \(y^4\), \(y\) is the base and 4 is the exponent. This means \(y \times y \times y \times y = y^4\).

In our exercise with terms \(y^4, y^5, y^{10}\), each term shares the same base \(y\), but the exponents are different. To find the GCF, we primarily focus on these exponents.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can contain numbers, variables, and operation signs. In our example, \(y^4\), \(y^5\), and \(y^{10}\) are algebraic expressions because they contain a variable \(y\) raised to a power.

Simplifying algebraic expressions like these often involves factoring out common terms. By finding the GCF, we simplify the expression, making it easier to work with.

General Rule for GCF

Finding the GCF of terms with the same base but different exponents follows a simple rule. You just need to look at the exponents. For terms \(y^a, y^b,\text{ and } y^c\), the GCF is determined by the smallest exponent among \(a, b,\text{ and } c\).

Therefore, the general rule is to identify the smallest exponent and use it with the common base. For \(y^a, y^b,\text{ and } y^c\), the GCF will be \(y^{\text{min}(a, b, c)}\).

In our exercise, the smallest exponent is 4, so the GCF of \(y^4, y^5,\text{ and } y^{10}\) is \(y^4\).