Problem 60 - Algebra and Trigonometry

Question

\(47-72\) . Simplify the expression, and eliminate any negative exponent(s). $$ \frac{\left(2 v^{3} w\right)^{2}}{v^{3} w^{2}} $$

Step-by-Step Solution

Answer

The simplified expression is \(4v^3\).

Step 1: Simplify the Power Expression

First, simplify the numerator of the fraction: \( (2v^3w)^2 \). Use the power of a product property: \((ab)^n = a^n b^n\). Thus, \((2v^3w)^2 = 2^2(v^3)^2(w)^2 = 4v^6w^2\).

Step 2: Write Simplified Fraction

After simplification, we have the expression: \( \frac{4v^6w^2}{v^3w^2} \).

Step 3: Simplify the Fraction

Simplify the fraction by canceling common terms in the numerator and the denominator. For the \(v\) terms, \(v^6\) in the numerator and \(v^3\) in the denominator, apply the rule \( \frac{a^m}{a^n} = a^{m-n} \), which gives \( v^{6-3} = v^3 \). For the \( w \) terms, \(w^2\) cancels completely since it appears in both the numerator and the denominator with the same power.

Step 4: Final Expression

After canceling, the simplified expression is \( 4v^3 \).

Key Concepts

Power of a Product Property

When working with expressions that involve exponents, understanding the power of a product property can significantly simplify the process. The power of a product property is one of the exponent rules that states: if you have a product of terms raised to a power,

• \[ (ab)^n = a^n b^n \]

This means that each term in the product is individually raised to the power, making calculations more straightforward.
This rule is particularly useful in algebraic simplification. For example, when you have an expression like

• \( (2v^3w)^2 \), use this property to simplify it to \( 2^2 \cdot (v^3)^2 \cdot w^2 = 4v^6w^2 \).

By distributing the exponent to each part of the product, complex expressions become much more manageable.
Remember, the property applies strictly to products within the parentheses, so ensure you apply the exponent correctly to each factor.

Simplifying Rational Expressions

Simplifying rational expressions involves reducing expressions to their simplest form. This is accomplished by cancelling out common terms in the numerator and the denominator.
In our example, the fraction is

• \( \frac{4v^6w^2}{v^3w^2} \).

Both the numerator and the denominator share common terms, allowing for simplification. First, focus on the common terms observed: the term \( w^2 \) appears in both places. Since they have the same exponent value, they cancel each other out, leaving no \( w \) terms in the expression.
Next, address the \( v \) terms:

• The numerator has \( v^6 \), and the denominator has \( v^3 \).


• Divide the terms by subtracting exponents as per exponent rules to get \( v^{6-3} = v^3 \).

These steps effectively simplify the expression to \( 4v^3 \). By understanding which terms can cancel, you can easily simplify complex fractions.

Exponent Rules

Exponent rules are crucial mathematical tools used to simplify expressions involving powers. These rules make working with exponents straightforward and systematic.
Here are some vital exponent rules often used:

• Product of Powers: When multiplying like bases, add their exponents: \( a^m \cdot a^n = a^{m+n} \).
• Power of a Power: An exponent applied to another exponent multiplies them: \( (a^m)^n = a^{m \cdot n} \).
• Quotient of Powers: When dividing like bases, subtract the exponents: \( \frac{a^m}{a^n} = a^{m-n} \).
• Zero Exponent: Any base raised to the power of 0 is 1: \( a^0 = 1 \) for any non-zero \( a \).

In our exercise, the simplification process relied on the quotient of powers rule:

• \( v^6 \div v^3 \) which simplifies to \( v^{6-3} = v^3 \).

Applying these rules can simplify complicated expressions and solve algebraic problems efficiently. Mastering them is key to handling various levels of math coursework.