Problem 114 - Algebra and Trigonometry

Question

Simplify each exponential expression. Assume that variables represent nonzero real numbers. $$ \frac{\left(2^{-1} x^{-3} y^{-1}\right)^{-2}\left(2 x^{-6} y^{4}\right)^{-2}\left(9 x^{3} y^{-3}\right)^{0}}{\left(2 x^{-4} y^{-6}\right)^{2}} $$

Step-by-Step Solution

Answer

\[\frac{1}{16x^{2}y^{2}}\]

Step 1: Apply the power of a product rule

The power of a product rule states that the power of a product is equal to the product of its factors raised to the given power, individually. Applying this rule, the expression simplifies to: \[ \frac{(2^{-2} x^{6} y^{2})(4 x^{12} y^{-8})(1)}{(4 x^{-8} y^{-12})}\]

Step 2: Apply the Negative exponent rule

The Negative exponent rule states that negative exponents in the numerator go to the denominator and become positive, and vice versa. Thus, we obtain:\[\frac{2^{-2}x^{6}y^{2} \cdot 1}{4x^{-8}y^{-12} \cdot 4x^{12}y^{-8}}\]

Step 3: Simplify each term

To simplify each term, combine all like terms using the law of exponents. The law of exponents states that when multiplying like bases, we add the exponents. This yields:\[\frac{2^{-2}x^{6}}{2^{2}x^{4}} \cdot \frac{y^{2}}{y^{4}} \cdot 1 = 2^{-4}x^{2}y^{-2}\]

Step 4: Apply negative exponent rule again

Applying the negative exponent rule to the final expression, we yield:\[\frac{1}{2^{4}x^{2}y^{2}}\]

Key Concepts

Understanding the Power of a Product Rule

The power of a product rule is a very handy tool when working with exponential expressions. This rule allows you to simplify expressions where a whole product is raised to a power.
Essentially, the rule says that you can distribute the exponent to each factor in the product.

• For example, \( (ab)^n = a^n \times b^n \). You apply the exponent \( n \) to each part of the product \( ab \).
• In the given problem, we use this rule to simplify terms like \( (2^{-1} x^{-3} y^{-1})^{-2} \).
• It becomes \( 2^{2} x^{6} y^{2} \), since each factor is raised to the power of \(-2\).

Once you apply this rule, your expression looks much simpler, and you can proceed with simplifying it further. Understanding how to properly use the power of a product rule can significantly ease the process of working with complex exponential expressions.

Deciphering the Negative Exponent Rule

Negative exponents can seem tricky at first, but once understood, they are quite straightforward.
The negative exponent rule indicates that a negative exponent essentially means taking the reciprocal of the base.

• In simpler terms, \( a^{-n} = \frac{1}{a^n} \).
• This means if you have a negative exponent in the numerator, it should be moved to the denominator and become positive.
• And vice versa — if it's in the denominator, it should go to the numerator.

In the problem, you see how negative exponents are manipulated: \( 2^{-2} \) becomes \( \frac{1}{2^2} \) when simplified correctly.
Knowing this rule helps immensely when simplifying complex expressions, as it transforms negative exponents into positive ones, making further operations much clearer.

Mastering the Law of Exponents

The law of exponents is the foundation upon which many simplifications are built.
These laws help in combining and breaking down exponential expressions.

• One key law is when multiplying like bases, you add the exponents: \( a^m \times a^n = a^{m+n} \).
• This is particularly useful when reducing complex fractions or multiplying factors with exponents.

In our exercise, this law allows for combining terms with like bases, simplifying from \( x^{6} \times x^{-8} \) to \( x^{-2} \).
Also, it makes it possible to simplify more efficiently by focusing only on the exponents instead of the entire expression.
By mastering this law, you set a solid ground for dealing with more elaborate expressions effortlessly.