Question

Simplify each expression. Express the answer so that all exponents are positive. Whenever an exponent is 0 or negative, we assume that the base is not \(0 .\) \(\left(\frac{2 x^{-3}}{3 y^{-1}}\right)^{-2}\)

Short answer

\frac{9 x^{6}}{4 y^{2}}

Step-by-step solution

Apply the Negative Exponent Rule

The expression \(\frac{2 x^{-3}}{3 y^{-1}}\)^{-2} can be simplified by first applying the negative exponent rule, which states \((a/b)^{-n} = (b/a)^{n}\). Applying this rule gives: \(\frac{3 y^{-1}}{2 x^{-3}} \)^{2}.

Simplify Inside the Parentheses

Next, simplify within the parentheses. Move the variables with negative exponents to the opposite denominator or numerator, making them positive: \(\frac{3 \times x^{3}}{2 \times y} \)^{2}.

Apply the Exponent to Both Numerator and Denominator

Now, apply the exponent of 2 to both numerator and denominator separately: \[ \frac{(3 x^{3})^2}{(2 y)^2} \]

Simplify Each Part

Simplify each part: \((3 x^{3})^{2} = 9 x^{6}\) and \((2 y)^{2} = 4 y^{2}\).

Write the Final Expression

Put it all together to get the simplified expression: \(\frac{9 x^{6}}{4 y^{2}} \).

Key concepts

Negative Exponent Rule

In algebra, handling negative exponents correctly is crucial. Let's remember the rule: an expression with a negative exponent, like \( a^{-n} \), can be rewritten as \( \frac{1}{a^n} \). For example, \( x^{-3} \) becomes \( \frac{1}{x^3} \).
This can be particularly useful when simplifying fractions with variables in the numerator and denominator.
When you see a fraction raised to a negative exponent, such as \(\frac{a}{b} \)^{-n}, you can flip the fraction and change the sign of the exponent to get \( \frac{b}{a} \)^{n}. This is because raising a fraction to a negative power essentially inverts the fraction and then raises it to the positive power.
Applying this rule to our exercise, we get: \(\frac{3y^{-1}}{2x^{-3}} \)^{2}.
Notice how we swapped the numerator and denominator due to the negative exponent of -2.

Positive Exponents

Positive exponents are straightforward; they tell us how many times to multiply the base by itself. For instance, \( x^3 \) means \( x \) multiplied by itself three times: \( x \times x \times x \).
In our problem, after applying the negative exponent rule, we need to ensure all exponents are positive. This means moving any variable with a negative exponent to the opposite side of the fraction.

• If a variable in the numerator has a negative exponent, move it to the denominator and make the exponent positive.
• Conversely, if a variable in the denominator has a negative exponent, move it to the numerator.

For instance, \( y^{-1} \) in our problem moves to the denominator to become \( y \), and \( x^{-3} \) moves to the numerator to become \( x^{3} \).
This leaves us with \( \frac{3x^{3}}{2y} \).

Simplification Steps

Simplifying algebraic expressions involves breaking down the problem step-by-step. Here are the clear steps for our exercise:
1. **Apply the Negative Exponent Rule:** Flip the fraction and change the sign of the exponent.
\( \frac{2 x^{-3}}{3 y^{-1}} \)^{-2} becomes \( \frac{3y^{-1}}{2x^{-3}} \)^2.
2. **Move Variables with Negative Exponents:** Make negative exponents positive by moving them to the other side of the fraction. \( \frac{3x^{3}}{2y} \).
3. **Apply the Exponent to Each Part:** Raise both numerator and denominator to the given power. \( \frac{(3 x^{3})^2}{(2 y)^2} \).
4. **Simplify:** Calculate the power for each term separately.

• For the numerator: \( (3 x^3)^2 = 9 x^6 \).
• For the denominator: \( (2 y)^2 = 4 y^2 \).

Putting these together, the final expression is \( \frac{9 x^6}{4 y^2} \).
Remember, practice these steps to make simplifying algebraic expressions much easier!