Problem 18 - Intermediate Algebra

Question

Simplify. Write answers using positive exponents. \(\frac{63 a^{4}}{81 a^{6} b^{3}}\)

Step-by-Step Solution

Answer

\( \frac{7}{9 a^2 b^3} \)

Step 1: Simplify the Coefficients

The coefficient part of the expression is \( \frac{63}{81} \). We can simplify it by dividing both the numerator and the denominator by their greatest common divisor, which is 9. Thus, \( \frac{63}{81} = \frac{63 \div 9}{81 \div 9} = \frac{7}{9} \).

Step 2: Simplify the Exponent of \(a\)

For the variable \(a\), we have \(a^4\) in the numerator and \(a^6\) in the denominator. Using the power of the quotient rule \( \frac{a^m}{a^n} = a^{m-n} \), we calculate: \( a^{4-6} = a^{-2} \).

Step 3: Simplify the Exponent of \(b\)

For the variable \(b\), \(b^3\) exists only in the denominator, and none in the numerator, equivalent to \( b^0 \). Thus, applying the rule of division of exponents: \( b^{0-3} = b^{-3} \).

Step 4: Write the Final Expression Using Positive Exponents

Combine the simplified coefficients and exponents from Steps 1, 2, and 3. We have \( \frac{7}{9} a^{-2} b^{-3} \). To express using only positive exponents, rewrite \(a^{-2}\) and \(b^{-3}\) in the denominator: \( \frac{7}{9 a^2 b^3} \).

Key Concepts

Simplifying Expressions

When simplifying mathematical expressions, the goal is to rewrite them in a simpler or more compact form without changing their value. This often involves reducing coefficients (numbers in front of variables), combining like terms, or simplifying exponentials. In our exercise, the expression to simplify was \( \frac{63 a^{4}}{81 a^{6} b^{3}} \).

• Start by looking at the numerical part: \( \frac{63}{81} \). Find the greatest common divisor, which is 9 in this case, and divide both the numerator and the denominator by it. So, \( \frac{63}{81} \) simplifies to \( \frac{7}{9} \). This process is about making the numbers in the fraction as small as possible, using whole numbers.
• Then, examine each variable separately. Use the laws of exponents to simplify the expressions. For instance, with \( a \), it's about comparing the exponents in the numerator and denominator and subtracting them.

Remember, simplifying expressions gives clearer insight into the mathematical relationships involved. It makes complex calculations easier and results more readable.

Positive Exponents

Using positive exponents is often preferred for clarity and simplicity. An exponent tells us how many times to multiply a base by itself. A positive exponent indicates regular multiplication.In the original problem, the expression \( a^{-2} b^{-3} \) involved negative exponents, which can seem confusing. Negative exponents indicate the reciprocal of the base raised to the positive version of the exponent. In simpler terms:

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

The conversion from negative to positive exponents simplifies the final expression to \( \frac{7}{9 a^2 b^3} \). This step ensures clarity and allows for easier computation in further calculations. Positive exponents help in understanding the size and growth of the values involved.

Power of the Quotient Rule

The Power of the Quotient Rule is a key concept when dealing with exponents. This rule states that when you divide two powers with the same base, you can subtract the exponents: \[ \frac{a^m}{a^n} = a^{m-n} \]Applying this rule effectively simplifies complex expressions. In this exercise:

• For \( a \), compare the powers of \( a \) in the numerator and the denominator: \( a^{4} \) and \( a^{6} \). According to the rule, \( a^{4-6} = a^{-2} \).
• For \( b \), since there is no \( b \) in the numerator (effectively \( b^0 \)), the result is \( b^{0-3} = b^{-3} \).

Understanding and applying the Power of the Quotient Rule allows for accurate simplification and helps in handling algebraic expressions involving exponents efficiently. This makes it a powerful tool in mathematics, particularly in algebra and calculus.