Problem 52 - Algebra and Trigonometry

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 the expression \(\frac{35 a^{14} b^{6}}{-7 a^{7} b^{3}}\) is \(-5 a^7 b^3\).

Step 1: Divide the Constants

Divide \(35\) by \(-7\). This results in \(-5\).

Step 2: Apply the Quotient of Powers Rule for the a Term

To simplify \(a^{14} / a^{7}\), subtract the exponent of the denominator from the exponent of the numerator, which gives \(a^{14-7} = a^{7}\).

Step 3: Apply the Quotient of Powers Rule for the b Term

To simplify \(b^{6} / b^{3}\), subtract the exponent of the denominator from the exponent of the numerator, which gives \(b^{6-3} = b^{3}\).

Key Concepts

Quotient of Powers Rule

When it comes to simplifying exponential expressions, understanding the quotient of powers rule is a must. This rule states that when you divide two powers with the same base, you can subtract the exponent of the denominator from the exponent of the numerator to find the power of the quotient. In other words, \(\frac{a^{m}}{a^{n}} = a^{m - n}\).For example, if we take the expression \(\frac{a^{14}}{a^{7}}\), according to the quotient of powers rule, we simply subtract the exponent in the denominator from the exponent in the numerator: \(a^{14-7} = a^{7}\). Just like this, long and seemingly complex expressions can be quickly and easily simplified, which is a great skill to have when working through algebra problems.

Exponents

Exponents play a crucial role in algebra, as they represent how many times a number, known as the base, is multiplied by itself. An expression like \(a^{n}\) tells you that 'a' is multiplied by itself 'n' times. For instance, \(a^{3}\) would mean \(a \times a \times a\). Understanding the basic properties of exponents, such as how to multiply and divide powers with the same base or how to raise a power to another power, is fundamental to algebraic simplification.

Basic Rules of Exponents

• Product of powers rule: \(a^{m} \times a^{n} = a^{m+n}\)
• Power of a power rule: \( (a^{m})^{n} = a^{m \times n}\)
• Zero exponent rule: \(a^{0} = 1\) for any non-zero 'a'

Using these rules, we can handle most expressions that contain exponents, leading to a deeper understanding of how to manipulate and simplify complex algebraic expressions.

Algebraic Simplification

Algebraic simplification is all about making expressions easier to understand and work with. This process typically involves combining like terms, applying the distributive property, and using rules for exponents, among others. The goal is to reduce an expression to its simplest form without changing its value.

Techniques for Simplification

• Combining like terms by adding or subtracting coefficients
• Applying the distributive property: \(a(b+c) = ab+ac\)
• Simplifying expressions involving exponents using rules like the quotient of powers
• Factoring expressions to reveal simpler multiplicative components

It's a bit like decluttering a room; we take an expression with many parts and break it down into the most straightforward version possible. This can make understanding the expression easier, and it's an essential process for solving algebra problems effectively. The step by step solution provided to our initial exercise is a practical application of algebraic simplification, combining the division of constants with the quotient of powers rule to streamline the expression.