Problem 37 - Algebra 2

Question

Simplify. Assume that no variable equals 0. $$ \left(\frac{4 x^{-3} y^{2}}{x y^{-5}}\right)^{-2} $$

Step-by-Step Solution

Answer

The simplified expression is \(\frac{x^8}{16y^{14}}\).

Step 1: Simplify the Expression Inside the Parentheses

The expression is \( \frac{4x^{-3}y^2}{xy^{-5}} \). We start by simplifying the exponents in the numerator and the denominator separately. Combine the like terms:- For \(x\): \(x^{-3}\) in the numerator and \(x\) or \(x^1\) in the denominator. Applying the rules of exponents, we have \(x^{-3-1}=x^{-4}\).- For \(y\): \(y^{2}\) in the numerator and \(y^{-5}\) in the denominator. This simplifies to \(y^{2-(-5)}=y^{2+5}=y^{7}\).Now, the expression inside the parentheses becomes \(4x^{-4}y^7\).

Step 2: Apply the Negative Exponent Outside the Parentheses

The simplified expression is now \( (4x^{-4}y^7)^{-2} \). A negative exponent indicates a reciprocal, namely \( a^{-b} = \frac{1}{a^b} \). So, apply this to each term inside:- The coefficient \(4\) becomes \(\frac{1}{4^2}=\frac{1}{16}\).- The term \(x^{-4}\) becomes \((x^{-4})^{-2}=x^{8}\) because \((-4)\times(-2) = 8\).- The term \(y^7\) becomes \((y^7)^{-2}=y^{-14}\) because \(7\times(-2)=-14\).Thus, we have the expression \(\frac{1}{16}x^{8}y^{-14}\).

Step 3: Simplify the Expression with Positive Exponents

The expression is \( \frac{1}{16}x^{8}y^{-14} \). Convert \(y^{-14}\) into a positive exponent:- \(y^{-14}\) is equivalent to \(\frac{1}{y^{14}}\).So the simplified form is \( \frac{x^{8}}{16y^{14}} \).

Step 4: Conclusion

Finally, we get the fully simplified expression. Check thoroughly to ensure all operations and exponent rules were applied correctly. The final simplified expression is \( \frac{x^8}{16y^{14}} \).

Key Concepts

Negative Exponents

Negative exponents occur when an exponent is less than zero. This might seem confusing at first, but the rule is really simple. A negative exponent means taking the reciprocal of the base with a positive exponent. For instance, if you have \[ a^{-b} \] , this is equivalent to \[ \frac{1}{a^b} \].
Here’s how you can apply it:

• For \( 4^{-1} \) , you get \( \frac{1}{4} \)
• If you have \( x^{-3} \) , when converted, it becomes \( \frac{1}{x^3} \)
Knowing this simple rule helps you flip and transform negative exponents, making them much more manageable.
In our exercise, we used the negative exponent rule with \( x^{-4} \) and \( y^{-14} \) . We simply changed these into positive exponents by taking their reciprocals.

Rules of Exponents

The rules of exponents are guidelines that help you manipulate expressions involving powers. They're essential for simplifying expressions and solving equations efficiently.
Some of the key rules include:

• Multiplying Powers with the Same Base: \( a^m \times a^n = a^{m+n} \)
• Dividing Powers with the Same Base: \( \frac{a^m}{a^n} = a^{m-n} \)
• Power of a Power: \( (a^m)^n = a^{m \times n} \)
• Power of a Product: \( (ab)^n = a^n \times b^n \)
• Negative Exponent: \( a^{-n} = \frac{1}{a^n} \)

Let's see how these rules were applied in our original exercise:
- To handle the numerator and denominator separately, we used the division rule. So \( x^{-3} \) in the numerator and \( x \) in the denominator turned into \( x^{-4} \).
- We also transformed the power on a power by making use of \( (x^{-4})^{-2} \) which simplifies to \( x^8 \). This is the power of a power rule at work.

Simplifying Algebraic Expressions

Simplifying algebraic expressions involves reducing expressions to make them easier to work with. This includes handling coefficients and like terms, as well as applying all the established rules of exponents.
Here are a few steps to keep in mind:

• Combine like terms. For example, adding or subtracting terms with the same variable power.
• Use the exponent rules to reduce complexity.
• Convert negative exponents to positive exponents by using reciprocals.
• Ensure coefficients are simplified as well. For instance, taking \( 4^{-2} \) turns into \( \frac{1}{16} \).

In the exercise provided, we approached simplification by using these steps:
- We managed coefficients and worked with both negative exponents and reciprocal forms.
- We combined like terms, such as \( x^{-3} \) and \( x \), ensuring the expression was in its simplest form by the end.
In conclusion, simplification isn't just mechanical; it’s about recognizing which rules apply and thoughtfully applying them to reach the optimal result.