Problem 52 - College Algebra

Question

Simplify each exponential expression $$ \frac{35 a^{14} b^{6}}{-7 a^{7} b^{3}} $$

Step-by-Step Solution

Answer

The simplified form of \( \frac{35 a^{14} b^{6}}{-7 a^{7} b^{3}} \) is \( -5 a^{7} b^{3} \)

Step 1: Initial Expression

The problem begins with the expression \( \frac{35 a^{14} b^{6}}{-7 a^{7} b^{3}} \)

Step 2: Simplify Coefficients

Firstly, simplify the coefficients by dividing 35 by -7, which gives -5, so the expression becomes \( -5 \frac {a^{14} b^{6}}{a^{7} b^{3}} \)

Step 3: Apply law of exponents for division

For same bases, if being divided, the exponents subtract. Therefore, \(a^{14}/a^{7}\) becomes \(a^{14-7}\) = \(a^{7}\), and \(b^{6}/b^{3}\) becomes \(b^{6-3}\) = \(b^{3}\). So, the expression becomes -5*a^7*b^3.

Step 4: Final Simplified Expression

The simplified form of the given expression is now \( -5 a^{7} b^{3} \)

Key Concepts

Law of Exponents

The law of exponents is a fundamental principle in algebra that guides how to handle expressions involving powers. One specific rule, relevant to our problem, is when you divide two exponential expressions with the same base. In such cases, you subtract the exponent of the denominator from the exponent of the numerator.
For instance, with the bases "a" and "b" in our example, this rule can be easily applied:

• \(a^{14}/a^{7} = a^{14-7} = a^{7}\)
• \(b^{6}/b^{3} = b^{6-3} = b^{3}\)

This simple subtraction helps you simplify what might initially look complex, making it manageable. It's important to remember that this rule only works when the bases are the same and it's key in many algebraic processes.

Simplifying Expressions

Simplifying expressions is a crucial skill in algebra that often involves reducing an expression to its most basic form. In our example, simplifying began with reducing the coefficient and using the law of exponents.

Here's how it unfolds:

• First, handle the coefficients. For \( \frac{35}{-7} \), simply divide 35 by -7 to get -5.
• Next, apply exponent rules. This involves reducing powers of the same base, as you've seen above.

The process helps not only in problem-solving but in understanding the nature of algebraic expressions more deeply. Simplifying is a method that often reveals the true structure and simplicity within seemingly complicated expressions.

Algebraic Fractions

Algebraic fractions are expressions that contain numerators and denominators with algebraic terms. Similar to numerical fractions, the principles of fraction arithmetic apply.

When simplifying algebraic fractions:

• Simplify coefficients, which are the numerical parts, first.
• Apply the law of exponents as shown to handle terms with powers.

In our example, the original algebraic fraction \( \frac{35 a^{14} b^{6}}{-7 a^{7} b^{3}} \) is simplified by treating each part of the fraction individually—the coefficients separately from the algebraic parts. This practice ensures each component correctly contributes to the final simplified form, which is crucial for accurate algebraic manipulation.