Problem 133 - Elementary Algebra

Question

For the following problems, write each expression so that only positive exponents appear. $$ \left(x^{-3} y^{7}\right)^{-4} $$

Step-by-Step Solution

Answer

Question: Rewrite the expression \(\left(x^{-3} y^{7}\right)^{-4}\) with only positive exponents. Answer: \(\left(x^{-3} y^{7}\right)^{-4} = x^{12} \cdot \frac{1}{y^{28}}\)

Step 1: Apply the Power Rule

The power rule states that \((a^m)^n = a^{mn}\). We will apply the power rule to each term inside the parentheses. $$ \left(x^{-3} y^{7}\right)^{-4} = x^{-3 \cdot -4}y^{7 \cdot -4} $$

Step 2: Multiply the Exponents

Next, we will multiply the exponents to simplify the expression. $$ x^{-3 \cdot -4}y^{7 \cdot -4} = x^{12}y^{-28} $$

Step 3: Write with Positive Exponents

To rewrite an expression with a negative exponent, we can take its reciprocal. So, let's apply this method for the negative exponent of variable y: $$ x^{12} y^{-28} = x^{12} \cdot \frac{1}{y^{28}} $$ Now, our expression has only positive exponents: $$ \left(x^{-3} y^{7}\right)^{-4} = x^{12} \cdot \frac{1}{y^{28}} $$

Key Concepts

Power Rule

The power rule is a fundamental principle when dealing with exponents. It's expressed as \((a^m)^n = a^{mn}\). This rule is very useful because it allows us to simplify expressions by multiplying the exponents. Imagine stacking powers like stacking boxes - each layer represents an exponent. This method significantly reduces the complexity of dealing with powers.
For example, when you have an expression like \(\left(x^{-3} y^{7}\right)^{-4}\), the power rule helps us understand that you need to multiply the exponent of each individual variable within the parentheses by the outer exponent. This transforms the problem into something much simpler to handle. Applying the power rule here changes the expression to \(x^{-3 \cdot -4}y^{7 \cdot -4}\), setting the stage for the following simplifications.

Negative Exponents

Understanding negative exponents is crucial because they often intimidate students at first. A negative exponent indicates that instead of multiplying, you should think about division or the reciprocal. So, the trick is to flip the expression.

• \(a^{-n} = \frac{1}{a^n}\)

When faced with \(x^{-3 \cdot -4}y^{7 \cdot -4}\), the exponents were originally negative. After applying the power rule and simplifying, the expression becomes \(x^{12}y^{-28}\). Here, \(y\) still has a negative exponent. This indicates that it needs to be moved to the denominator to convert it back to a positive exponent.

Exponent Multiplication

Exponent multiplication is the process of combining powers through multiplication. When using exponent multiplication, it's essential to simplify expressions efficiently. Through using the power rule, we've already set up the problem for easier multiplication.In the expression \(x^{-3 \cdot -4}y^{7 \cdot -4}\), multiply the exponents directly:

• \(-3 \times -4 = 12\)
• \(7 \times -4 = -28\)

Thus we arrive at \(x^{12}y^{-28}\). This part is critical because getting the correct magnitude of each variable’s exponent ensures the problem transitions smoothly into the stage where positive exponents can be achieved.

Reciprocal

The concept of a reciprocal is straightforward yet powerful, especially when dealing with negative exponents. A reciprocal of a number is essentially 1 divided by that number. With exponents, the reciprocal helps convert negative exponents into positive ones.When you encounter a negative exponent like \(y^{-28}\), converting it to positive involves taking the reciprocal. Therefore, \(y^{-28}\) becomes \(\frac{1}{y^{28}}\).
This step is necessary in rewriting the expression with only positive exponents, which is often a requirement in algebraic problems. Therefore, the expression \(x^{12}y^{-28}\) turns into \(x^{12} \cdot \frac{1}{y^{28}}\), a format where all exponents are positive and neatly organized. Understanding and applying the reciprocal correctly is crucial for simplifying expressions in mathematics.