Problem 64 - 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. $$ \frac{15 x^{20} y^{24} z^{4}}{5 x^{19} y z} $$

Step-by-Step Solution

Answer

Question: Simplify the expression: $$\frac{15 x^{20} y^{24} z^{4}}{5 x^{19} y z}$$ Answer: The simplified expression is $$3 x y^{23} z^3$$.

Step 1: Identify the given expression

Here is the expression we need to simplify: $$ \frac{15 x^{20} y^{24} z^{4}}{5 x^{19} y z} $$

Step 2: Apply quotient rule of exponents for each variable

The quotient rule of exponents states that when dividing two expressions with the same base and exponents, we can subtract the exponents. We apply this rule to each variable in the expression: $$ \frac{x^{20}}{x^{19}} = x^{20-19} =x^1=x $$ $$ \frac{y^{24}}{y^1} = y^{24-1} =y^{23} $$ $$ \frac{z^4}{z^1} = z^{4-1} =z^{3} $$

Step 3: Simplify the numerical part of the expression

Since we have a numerical fraction, we have to simplify it: $$ \frac{15}{5} = 3 $$

Step 4: Combine all simplified parts

Finally, we combine all the simplified parts: $$ \frac{15 x^{20} y^{24} z^{4}}{5 x^{19} y z} = 3 \cdot x \cdot y^{23} \cdot z^3 $$ So the simplified expression is: $$ 3 x y^{23} z^3 $$

Key Concepts

Product Rule

In mathematics, the product rule is a fundamental concept when dealing with exponents. It helps us to simplify expressions where exponents are involved. When multiplying two expressions with the same base, the product rule states that we add the exponents. For example, for any non-zero number, the expression \(a^m \times a^n = a^{m+n}\). This rule easily allows us to condense large exponent terms.
When applying the product rule, it's important to have variables with the same base. Suppose our expression is \(2^3 \times 2^4\). Here, the base 2 is consistent, and applying the product rule gives \(2^{3+4} = 2^7\). This approach helps in quickly reducing the complexity of problems involving exponents.

Quotient Rule

The quotient rule for exponents is another useful tool in algebra. It assists in simplifying expressions that involve division of like bases. According to this rule, when dividing one exponential expression by another with the same base, you subtract the exponent of the denominator from the exponent of the numerator.
In mathematical terms, for non-zero base \(a\), the quotient rule is \(\frac{a^m}{a^n} = a^{m-n}\). For example, if we have the expression \(\frac{x^7}{x^5}\), the quotient rule helps us simplify it to \(x^{7-5} = x^2\). This process reduces the complexity of expressions significantly.

• It is crucial to ensure the bases are the same; otherwise, this rule cannot be applied directly.
• The rule covers circumstances with whole number exponents.
• Always check if any resulting negative exponents need further simplification by rewriting them in positive form.

In our given problem, understanding the quotient rule allowed us to simplify each term efficiently, directly impacting the simplicity of the result.

Simplifying Expressions

Simplifying expressions with exponents is a core skill in algebra that enhances mathematical understanding and problem-solving abilities. It involves reducing expressions to their simplest form using rules like the product rule and quotient rule, along with basic arithmetic operations.
The process of simplification usually follows a few straightforward steps:

• Apply relevant exponent rules: Use the product and quotient rules as needed to simplify the powers.
• Simplify any coefficients: Perform basic arithmetic operations on the numerical parts of the expression.
• Combine and organize: Once each part is simplified, combine them to form the simplest possible expression.

In the example exercise given, we simplified \(\frac{15 x^{20} y^{24} z^{4}}{5 x^{19} y z}\) by first using the quotient rule to handle the variables with exponents, and then simplified the coefficients by dividing 15 by 5.
Finally, the expression \(3 x y^{23} z^3\) is the most simplified form, ensuring clarity and ease of understanding. Simplifying not only helps in computations but also in recognizing underlying patterns.