Problem 52 - Beginning and Intermediate Algebra

Question

Simplify. Assume all variables represent nonzero real numbers. The answer should not contain negative exponents. $$\left(\frac{10 x^{-5} y}{20 x^{5} y^{-3}}\right)^{-2}$$

Step-by-Step Solution

Answer

The simplified expression is \(y^8\).

Step 1: Rewrite fraction as individual exponents and base

Start by rewriting the fraction raised to a negative power as individual factors raised to that power: \[\left(\frac{10 x^{-5} y}{20 x^{5} y^{-3}}\right)^{-2} = \frac{(10 x^{-5} y)^{-2}}{(20 x^{5} y^{-3})^{-2}}\]

Step 2: Distribute power to each factor inside parentheses

Now, distribute the -2 power to each factor inside the parentheses: \[\frac{(10 x^{-5} y)^{-2}}{(20 x^{5} y^{-3})^{-2}} = \frac{10^{-2}(x^{-5})^{-2}(y)^{-2}}{20^{-2}(x^5)^{-2}(y^{-3})^{-2}}\]

Step 3: Apply exponent rules to simplify exponents

Apply exponent rules: - For negative powers, use the rule \(a^{-n} = \frac{1}{a^n}\). - For powers raised to another power, use the rule \((a^n)^m = a^{nm}\). \[\frac{10^{-2}(x^{-5})^{-2}(y)^{-2}}{20^{-2}(x^5)^{-2}(y^{-3}){-2}} = \frac{\frac{1}{10^2}\left(\frac{1}{x^{-10}}\right)\frac{1}{y^2}}{\frac{1}{20^2}\left(\frac{1}{x^{-10}}\right)\frac{1}{y^{-6}}}\]

Step 4: Combine the factors with their corresponding base

Now, combine the factors with the same base in the numerator and denominator: \[\frac{\frac{1}{10^2}\left(\frac{1}{x^{-10}}\right)\frac{1}{y^2}}{\frac{1}{20^2}\left(\frac{1}{x^{-10}}\right)\frac{1}{y^{-6}}} = \frac{\frac{1}{10^2 x^{-10} y^2}}{\frac{1}{20^2 x^{-10} y^{-6}}}\]

Step 5: Cancel out common factors

Now, cancel out the common factors between the numerator and the denominator: \[\frac{\frac{1}{10^2 x^{-10} y^2}}{\frac{1}{20^2 x^{-10} y^{-6}}} = \frac{1}{\frac{10^2}{20^2} \cdot {\frac{x^{-10}}{x^{-10}}} \cdot \frac{y^2}{y^{-6}}} \]

Step 6: Simplify the remaining expression

Finally, simplify the remaining expression. - For the numerical part, divide \(10^2\) by \(20^2\). - For the y part, use the rule: \(\frac{y^a}{y^b} = y^{a-b}\) \(\frac{1}{\frac{10^2}{20^2} \cdot {\frac{x^{-10}}{x^{-10}}} \cdot \frac{y^2}{y^{-6}}}\) = \( y^{2-(-6)} = y^8\) The simplified expression is \(y^8\).

Key Concepts

Exponent Rules

Understanding exponent rules can greatly simplify complex algebraic expressions. Exponents are a way to express repeated multiplication of a number or variable by itself. There are a few basic rules that can help:

• Product of Powers Rule: If you multiply two powers with the same base, you add the exponents: \[a^m \cdot a^n = a^{m+n}\]
• Power of a Power Rule: To raise a power to another power, you multiply the exponents: \[(a^m)^n = a^{m \cdot n}\]
• Power of a Product Rule: The power of a product is the product of the powers: \[(ab)^m = a^m \cdot b^m\]
• Quotient of Powers Rule: For division, you subtract the exponents: \[\frac{a^m}{a^n} = a^{m-n}\]

These rules make it easier to simplify expressions, like the one in our example by systematically applying them to rewrite and eventually reduce the expression.

Negative Exponents

Negative exponents can seem confusing at first, but they are quite straightforward once you grasp the core idea. A negative exponent indicates that the base should be taken as the reciprocal raised to the corresponding positive exponent. The rule is simple: \[a^{-n} = \frac{1}{a^n}\]
Applying this rule means a negative exponent reflects the expression across the fraction line. In our original exercise, the negative exponent \((x^{-5})^{-2}\) was transformed to a positive exponent: \[(x^5)^2 = x^{10}\]. Similarly, other negative powers were inverted by switching their sign and finding the corresponding positive exponent.
This straightforward application allows you to convert a problematic expression into something manageable. Importantly, never leave your final answers with negative exponents unless specified.

Fraction Simplification

Simplifying fractions involves both numerical and variable components when dealing with algebra. For numerical fractions, divide the top and bottom by their greatest common divisor to reduce the fraction. For example, in the expression \(\frac{10^2}{20^2}\), both numbers are divisible by \(10^2\), simplifying it to \(\left(\frac{1}{2}\right)^2 = \frac{1}{4}\).
When variables are involved, like \(\frac{y^2}{y^{-6}}\), you use the rules of exponents to simplify them. Subtract the exponent in the denominator from the exponent in the numerator: \[y^{2-(-6)} = y^8\].
Always ensure that your final answer is in the simplest form, with no further simplifications possible. Fraction simplification is a crucial step in reducing complex expressions to their simplest form.