Problem 69 - Algebra and Trigonometry

Question

\(47-72\) . Simplify the expression, and eliminate any negative exponent(s). $$ \left(\frac{y}{5 x^{-2}}\right)^{-3} $$

Step-by-Step Solution

Answer

The expression simplifies to \( \frac{125}{y^{3} x^{6}} \).

Step 1: Rewrite the Expression with Positive Exponents

Start with the given expression: \( \left(\frac{y}{5 x^{-2}}\right)^{-3} \). Notice that the term \( x^{-2} \) has a negative exponent. Rewrite it as \( \frac{1}{x^{2}} \). So the expression inside the parenthesis becomes \( \frac{y}{5} \times x^{2} \).

Step 2: Apply the Negative Exponent Outside the Parentheses

Now we deal with the \( -3 \) exponent outside the parentheses. Recall that a negative exponent indicates a reciprocal. Thus, \( \left(\frac{y}{5} \times x^{2}\right)^{-3} \) translates to \( \left(\frac{5}{y} \times \frac{1}{x^{2}}\right)^{3} \).

Step 3: Raise Each Component to the Power of 3

Raise each part within the parentheses to the power of 3: \( \left(\frac{5}{y} \right)^{3} \times \left(\frac{1}{x^{2}}\right)^{3} \). This results in \( \frac{5^{3}}{y^{3}} \times \frac{1}{x^{6}} \) because \( x^{2} \) raised to 3 is \( x^{6} \).

Step 4: Simplify the Result

Simplify the expression by multiplying across: \( \frac{5^{3}}{y^{3} x^{6}} = \frac{125}{y^{3} x^{6}} \).

Key Concepts

Negative Exponents

Negative exponents can often feel tricky, but they're not too complicated once you understand them. When you see a negative exponent, it means you need to take the reciprocal of the base.
For example, if you have a variable or number with a negative exponent, like \(x^{-2}\), you should rewrite it as \(\frac{1}{x^{2}}\). This process is often called "flipping" the base to the denominator.
So, rather than fearing negative exponents, just remember that the exponent tells you to "flip" the base. This perspective can make dealing with negative exponents a lot easier. Always aim to rewrite expressions to get rid of negative exponents. This will make subsequent calculations more manageable.

Simplifying Expressions

Simplifying expressions is a fundamental part of algebra. It's all about taking complex expressions and making them more concise or easier to work with.
In the given exercise, simplifying involved several steps:

• First, we converted any negative exponents to positive ones by taking their reciprocals.
• Next, we applied the outside exponent of \(-3\) to each component inside the parenthesis.
• Finally, we simplified each part separately and multiplied the terms.

Each step required careful attention to the properties of exponents such as applying powers to fractions.
The goal of simplifying is to achieve a form that might be more intuitive or manageable, particularly in preparation for additional algebraic manipulations. This process often helps in solving equations or comparing the sizes of algebraic terms.

Reciprocal of Exponents

The concept of reciprocal is essential when dealing with exponents, particularly negative ones. You learned earlier that a negative exponent like \(x^{-n}\) means you need the reciprocal, \(\frac{1}{x^{n}}\).
The same idea applies when an entire expression is raised to a negative exponent. For example, \( \left(\frac{a}{b}\right)^{-n} \) means you flip the fraction to be \( \left(\frac{b}{a}\right)^{n} \).
Understanding reciprocals is quite helpful because it allows you to convert uncomfortable negative exponents to positive ones. This makes computations simpler and less prone to mistakes.

• Think of reciprocals as a "flip" not only of the position but of the exponent's sign.
• It's a stepping stone toward unraveling more complex expressions.

Remember that the reciprocal serves as a bridge to rewrite expressions in a easily understandable and more conventional form.