Problem 47 - Precalculus Mathematics for Calculus

Question

Simplify the expression and eliminate any negative exponents(s). (a) \(\left(\frac{a^{2}}{b}\right)^{5}\left(\frac{a^{3} b^{2}}{c^{3}}\right)^{3}\) (b) \(\frac{\left(u^{-1} v^{2}\right)^{2}}{\left(u^{3} v^{-2}\right)^{3}}\)

Step-by-Step Solution

Answer

(a) \( \frac{a^{19} b^{11}}{c^{9}} \), (b) \( \frac{v^{10}}{u^{11}} \)."

Step 1: Distribute Exponents - Part (a)

First, we'll simplify each part of (a) separately by distributing the exponents within the parentheses. For the first term \( \left(\frac{a^{2}}{b}\right)^{5} \), \( a^{2} \) becomes \( a^{10} \) and \( b \) becomes \( b^{5} \). For the second term \( \left(\frac{a^{3} b^{2}}{c^{3}}\right)^{3} \), \( a^{3} \) becomes \( a^{9} \), \( b^{2} \) becomes \( b^{6} \), and \( c^{3} \) becomes \( c^{9} \).

Step 2: Combine Like Terms - Part (a)

Now, combine the like terms from the two simplified expressions. \( a^{10} \times a^{9} = a^{19} \) and \( b^{5} \times b^{6} = b^{11} \). Since there is no like term for \( c^{9} \), it remains as is: \( \frac{a^{19} b^{11}}{c^{9}} \).

Step 3: Distribute Exponents - Part (b)

For part (b), start by distributing the exponents. In the numerator \( \left(u^{-1}v^{2}\right)^{2} \), you apply the exponent as follows: \( u^{-2} \) and \( v^{4} \). In the denominator \( \left(u^{3}v^{-2}\right)^{3} \), apply the exponent to get \( u^{9} \) and \( v^{-6} \).

Step 4: Simplify Fractions and Eliminate Negative Exponents - Part (b)

In part (b), simplify by subtracting in the exponents for like bases. For \( u \): \( u^{-2} - u^{9} = u^{-11} \). For \( v \): \( v^{4} - v^{-6} = v^{10} \). Now, eliminate any negative exponents by rewriting \( u^{-11} \) as \( \frac{1}{u^{11}} \). The final expression is \( \frac{v^{10}}{u^{11}} \).

Key Concepts

Negative Exponents

Handling negative exponents can initially seem tricky, but they follow straightforward rules. A negative exponent indicates division, essentially meaning that the base should be on the opposite side of the fraction. For example, if you have \( u^{-11} \), this is the same as \( \frac{1}{u^{11}} \). Negative exponents allow expressions to be rewritten without them, making it easier to simplify further. Keep this simple conversion rule in mind, and you'll find negative exponent problems much more manageable.
Understanding this concept aids in cleaning up expressions and attaining a form that's easier to work with.

Simplifying Expressions

Simplifying expressions means making them more manageable or condensed while retaining the original values. It's all about breaking down complex terms into more straightforward forms. Generally, this includes distributing exponents, as seen in our original exercise, and dealing with negative exponents appropriately.

• First, handle each set of exponents separately within their parentheses.
• Then, rewrite any negative exponents to positive by flipping their base across the fraction bar.

By these changes, you maximize the simplicity of expressions, reducing complex fractions or terms into smaller, equivalent forms that are much easier to handle mathematically.

Distributive Property

The distributive property is crucial in handling expressions with exponents. It states that you can multiply the exponent outside of parentheses by each of the exponents inside. For instance, in the expression \( (a^2)^{5} \), you apply the property to get \( a^{10} \) since \( 2 \times 5 = 10 \).
This property extends to entire fractions as well, where every numerator and denominator's exponent gets multiplied by this outer exponent.

• This step ensures each term in the expression is raised correctly making combining terms easier later.

Mastering the distributive property with exponents is key to tackling advancement in algebra.

Combining Like Terms

Combining like terms optimizes expressions further by grouping powers of the same base together. When having multiple terms like \( a^{10} \) and \( a^{9} \), add their exponents to form \( a^{19} \), consolidating into a single power term. The process involves:

• Identifying similar bases.
• Adding or subtracting their exponents accordingly based on multiplication or division.

This results in streamlined expressions, removing redundancy and preparing the expression for final evaluation or further manipulation. By grouping and simplifying, computations become straightforward, preventing errors down the line.