Problem 41 - Elementary Algebra

Question

Use the product rule and quotient rule of exponents to simplify the following problems. Assume that all bases are nonzero and that all exponents are whole numbers. $$ 4 y^{4} \cdot 5 y^{6} $$

Step-by-Step Solution

Answer

Question: Simplify the expression \(4y^4 \cdot 5y^6\). Answer: The simplified expression is \(20y^{10}\).

Step 1: Write down the given expression

The given expression is: $$ 4y^4 \cdot 5y^6 $$

Step 2: Apply product rule to the exponents

According to the product rule, when multiplying two expressions with the same base, the exponents are added together. In this case, the base is \(y\). Adding the exponents we have: $$ y^{4 + 6} = y^{10} $$

Step 3: Multiply the coefficients

Now we need to multiply the coefficients 4 and 5: $$ 4 \cdot 5 = 20 $$

Step 4: Combine the coefficients and the exponential expression with the new exponent

Combine the result from Steps 2 and 3 to get the final simplified expression: $$ 20y^{10} $$ So, the simplified expression is \(20y^{10}\).

Key Concepts

Product Rule of Exponents

When you come across multiplication involving exponential terms with the same base, you can apply the product rule of exponents to simplify the expression. This rule states that when you multiply expressions with the same base, you keep the base and add the exponents. For example, if you have \( a^m \times a^n \), the product rule tells us that the result is \( a^{m+n} \).

Let's put this into practice with the given problem from the exercise: \( 4y^4 \times 5y^6 \). Both terms have \(y\) as the base, and by the product rule, we add the exponents 4 and 6 to get \( y^{4+6} = y^{10} \), resulting in the exponent part of the simplified expression.

Quotient Rule of Exponents

The quotient rule of exponents comes into play when you're dividing exponential expressions with the same base. According to this rule, you subtract the exponent of the denominator from the exponent of the numerator. The formula is \( \frac{a^m}{a^n} = a^{m-n} \), where \( a \) is a nonzero base.

Even though this problem does not require the quotient rule, it's vital to understand its application. If we had a problem like \( \frac{4y^6}{2y^2} \), we would subtract the exponents, which would result in \( 4y^{6-2} \). After simplifying, it would look like \( 2y^4 \).

Exponential Expressions

Exponential expressions are a way to express repeated multiplication compactly. An exponential expression consists of a base and an exponent. The base is the number being multiplied, and the exponent tells us how many times the base is used as a factor.

In the expression \( 20y^{10} \), \(y\) is the base, and 10 is the exponent, indicating that \(y\) is multiplied by itself 10 times. Understanding how to read and interpret these expressions is essential when working with algebraic operations involving exponents.

Algebraic Simplification

Algebraic simplification is the process of making an algebraic expression easier to understand and work with by combining like terms, using exponent rules, and performing arithmetic operations. Simplification can help in solving equations, evaluating algebraic expressions, and can provide clearer insights into the problem you are trying to solve.

For instance, in \( 4y^4 \times 5y^6 \), we simplify by combining like terms. This involves multiplying the numeric coefficients (4 and 5) and then applying the product rule to the exponential terms with the same base \(y\). The result is a much simpler expression, \( 20y^{10} \), that is easier to utilize in further calculations or equations. Understanding how to simplify expressions is a fundamental skill in algebra that allows students to navigate more complex problems effectively.