Problem 38 - Beginning and Intermediate Algebra

Question

Simplify. $$\frac{20 a^{5} b^{-2}}{5 a^{7} b^{-3}}$$

Step-by-Step Solution

Answer

The simplified expression is \(4 a^{-2} b\).

Step 1: Divide the numerical coefficients

Divide the numbers 20 and 5. \(\frac{20}{5} = 4\)

Step 2: Divide the variables with the same base (a)

In the given fraction, we have \(a^{5}\) in the numerator and \(a^{7}\) in the denominator. Use the quotient rule for exponents, which states that if you divide two powers with the same base, you subtract the exponents. So for this part, we should compute \(a^{5} / a^{7}\), which can also be written as \(a^{5-7}\), which simplifies to \(a^{-2}\).

Step 3: Divide the variables with the same base (b)

In the given fraction, we have \(b^{-2}\) in the numerator and \(b^{-3}\) in the denominator. Use the quotient rule for exponents, which states that if you divide two powers with the same base, you subtract the exponents. So for this part, we should compute \(b^{-2} / b^{-3}\), which can also be written as \(b^{-2 - (-3)}\), which simplifies to \(b^{1}\) or simply \(b\).

Step 4: Combine the results

Combine the results from steps 1-3 to form the simplified expression. We have: - Numerical coefficient: \(4\) - Simplified variable a: \(a^{-2}\) - Simplified variable b: \(b\) Therefore, the simplified expression is \(4 a^{-2} b\).

Key Concepts

Quotient Rule for Exponents

The quotient rule for exponents is a fundamental principle in algebra that helps simplify expressions involving powers of the same base. When dividing powers, keep the base and subtract the exponents.
For example, if you have an expression like \( \frac{a^m}{a^n} \), the quotient rule states that it simplifies to \( a^{m-n} \).
This simple operation makes handling complex algebraic expressions easier and is widely applicable in algebra and beyond.

• This rule applies only if the bases are the same, regardless of whether those bases are numbers or variables.
• It is essential to note the sign of the exponents as you subtract them. For example, subtracting a negative exponent effectively adds the opposite.
  e.g., \( b^{-2}/b^{-3} = b^{-2 - (-3)} = b^{1} \), which simplifies to \( b \).

Simplifying Expressions

Simplifying expressions is about making them as concise as possible without changing their value.
This process often involves applying rules like the quotient rule for exponents but may encompass other algebraic operations as well.

• Simplification often entails reducing numerical coefficients by dividing them, as seen in dividing 20 by 5 to obtain 4 in the initial expression.
• It involves handling each part of the expression separately: numerical parts, each variable, and their exponents.
• Expanding, factoring, and combining like terms are other possible simplification methods. In practice, simplification aids in revealing equivalent forms of expressions that are more practical for further calculations.

Exponents in Algebra

Exponents are a critical component of algebra that describe how many times a base number is multiplied by itself.
In algebra, understanding how to manipulate exponents is fundamental to simplifying expressions and solving equations.

• An exponent can be positive, negative, or zero, each offering a different meaning. For example, a negative exponent, like \( a^{-n} \), represents \( \frac{1}{a^n} \), a concept essential for simplification.
• Remember that any expression raised to the zero power is 1. For instance, \( b^0 = 1 \).
• Handling exponents involves applying rules such as the power rule, the product rule, and indeed the quotient rule, all of which help streamline algebraic work.
• Consequently, they enable the simplification and solution of various algebraic expressions, essential for confidence in solving more advance problems.