Problem 58 - Elementary Algebra

Question

Use the power rules for exponents to simplify the following problems. Assume that all bases are nonzero and that all variable exponents are natural numbers. $$ \frac{\left(9 a^{4} b^{0}\right)\left(2 b^{2} e\right)}{\left(3 a^{3} b\right)(6 b c)} $$

Step-by-Step Solution

Answer

Question: Simplify the given expression: $$ \frac{\left(9 a^{4} b^{0}\right)\left(2 b^{2} e\right)}{\left(3 a^{3} b\right)(6 b c)} $$ Answer: The simplified expression is: $$ abe $$

Step 1: Simplify the constants

To simplify the constants, we can multiply the constants in the numerator and denominator separately. Numerator: $$ 9 \times 2 = 18 $$ Denominator: $$ 3 \times 6 = 18 $$ Now we can cancel out the constants: $$ \frac{18}{18} = 1 $$

Step 2: Simplify the variables with exponents

Now we need to simplify the variables with exponents. We will use the product rule, which states that for any non-negative integers m and n, and any nonzero number a, we have: \(a^m \times a^n = a^{m+n}\) Numerator: We have \(a^4\) and \(b^0\). Since any nonzero number raised to the power of 0 is 1, we can ignore \(b^0\). Denominator: We have \(a^3\) and \(b\). Also, in the numerator, we have \(b^2\). So we can write the expression as: $$ \frac{a^4 \times b^2}{a^3 \times b} $$ Now let's simplify the expression using the quotient rule where \(a^{m-n}\)\ so, $$ {\displaystyle \frac {a^{4}\times b^{2}}{a^{3}\times b} =a^{4-3} \times b^{2-1}} $$

Step 3: Combine the simplified constants and variables

We already found that constants simplify to 1. And from Step 2, we have \(a^{4-3} \times b^{2-1}\). Now let's simplify this expression: $$ a^{4-3} \times b^{2-1} = a^1 \times b^1 = ab $$ Finally, the simplified expression is: $$ \frac{\left(9 a^{4} b^{0}\right)\left(2 b^{2} e\right)}{\left(3 a^{3} b\right)(6 b c)} = abe $$

Key Concepts

Product Rule for Exponents

The product rule for exponents is a useful tool in algebra. It helps simplify expressions where the same base is raised to different exponents. The rule states that when multiplying two expressions with the same base, you simply add their exponents. Mathematically, it is shown as:

• \( a^m \times a^n = a^{m+n} \)

This rule works because multiplication is repeated addition. For example, if we have \( a^4 \times a^5 \), it becomes \( a^{4+5} = a^9 \).
In practical terms, consider the expression \( x^2 \times x^3 \). Using the product rule, you add the exponents (2 and 3) to simplify: \( x^{2+3} = x^5 \).
This process can make complex algebraic expressions easier to handle by reducing the number of terms.

Quotient Rule for Exponents

The quotient rule for exponents is essential for dividing expressions with the same base. When dividing two expressions with identical bases, you subtract the exponent in the denominator from the exponent in the numerator. The mathematical expression is:

• \( \frac{a^m}{a^n} = a^{m-n} \)

This subtractive process simplifies the expression, making it easier to solve. For example, \( \frac{x^7}{x^3} \) simplifies to \( x^{7-3} = x^4 \).
It's important to remember that this rule only works when the base is the same and the base itself is non-zero. In terms of the original problem, we applied the rule:

• \( \frac{a^4}{a^3} = a^{4-3} = a^1 \)
• \( \frac{b^2}{b^1} = b^{2-1} = b^1 \)

This simplification plays a crucial role in ensuring the final expressions are neat and elegant.

Simplifying Algebraic Expressions

Simplifying algebraic expressions involves rewriting them in their simplest form, making them easier to understand and work with.
One of the primary goals of simplification is to reduce complexity while retaining the same value. The use of exponent rules, like product and quotient rules, allows us to achieve this effectively.
In the provided exercise, we began by simplifying constants: \( \frac{18}{18} = 1 \), eliminating unnecessary terms.
Next, we utilized the product and quotient rules for exponents to combine and reduce the power of variables. Simplifying expressions means eliminating redundant parts, such as a variable raised to zero power, which equals one, thus having no impact on the expression.

• After simplification, every term is expressed in its minimal form.
• The expression becomes more efficient to use in further calculations.

Simplification is a valuable method for solving equations and tackling complex algebraic problems, enabling clearer insight into the relationships between different parts of an equation.