Problem 29 - Elementary Algebra

Question

Simplify and write \(\left(x^{-4} y^{5}\right)^{-3}\left(x^{-6} y^{4}\right)^{2}\) so that only positive exponents appear.

Step-by-Step Solution

Answer

Question: Simplify the expression \(\left(x^{-4} y^{5}\right)^{-3} \cdot \left(x^{-6} y^4\right)^2\) with only positive exponents. Answer: \(\frac{1}{y^7}\)

Step 1: Apply exponent rules to each term

First, we need to apply the exponent rules to each term separately. Let's focus on the first term \(\left(x^{-4} y^{5}\right)^{-3}\). According to the exponent rule \((a^b)^c = a^{bc}\): \((x^{-4})^{-3} = x^{(-4)(-3)} = x^{12}\) and \((y^5)^{-3} = y^{(5)(-3)} = y^{-15}\) Now let's apply the same rules to the second term \(\left(x^{-6} y^4\right)^2\): \((x^{-6})^2 = x^{(-6)(2)} = x^{-12}\) and \((y^4)^2 = y^{(4)(2)} = y^8\)

Step 2: Combine the simplified terms

Now that we have simplified each term, let's combine the terms: \((x^{12}y^{-15})(x^{-12}y^8)\)

Step 3: Apply the product of powers rule

Now, we apply the product of powers rule, which states that \(a^m a^n = a^{m+n}\): \(x^{12}x^{-12}\) will give \(x^{12-12}\), which equals \(x^0\) \(y^{-15}y^8\) will give \(y^{-15+8}\), which equals \(y^{-7}\) So the expression becomes: \((x^0)(y^{-7})\)

Step 4: Simplify the expression with only positive exponents

As any number to the power of 0 is equal to 1, \(x^0 = 1\). Thus, we need to change the negative exponent of y to a positive one. We know that \(a^{-n} = \frac{1}{a^n}\): \(y^{-7} = \frac{1}{y^7}\) So, the final simplified expression with only positive exponents is: \(1 \cdot \frac{1}{y^7}\) or simply \(\frac{1}{y^7}\)

Key Concepts

Simplifying Expressions

Simplifying mathematical expressions might appear tricky at first, but it's all about making complex expressions as straightforward as possible. When we simplify an expression, we are typically trying to reduce it to its simplest form without losing any of its original value. We do this by applying arithmetic operations and rules that allow us to combine similar terms.

For expressions involving exponents, simplification often involves understanding and applying exponent rules like the power rule, product rule, or combining like terms. These rules help us reduce the number of operations or terms in the expression. In our exercise, simplifying was all about getting rid of negative exponents and combining terms in a way that only positive exponents remain at the end.

It's essential to know these rules well, as they are the backbone of working with expressions in algebra. Once you are familiar with the basic exponent rules, simplification becomes much more manageable.

Negative Exponents

Negative exponents might seem daunting, but they just indicate an inverse relationship. Essentially, a negative exponent tells you to take the reciprocal of the base, and then apply the positive exponent. In simpler terms, for any non-zero number, the rule is:

• \(a^{-n} = \frac{1}{a^n}\)

This means that \(a^{-n}\) is equivalent to the fraction \(\frac{1}{a^n}\), turning the negative exponent into a positive by moving the base to the denominator.

In practical terms, what this means for our exercise is taking terms like \(y^{-7}\) and rewriting them as \(\frac{1}{y^7}\). This transformation is crucial because many problems ask for expressions to only have positive exponents. Understanding this reversal makes working with negative exponents more accessible.

Product of Powers

The product of powers rule is a key exponent rule that allows for combining terms with the same base. This rule states that when multiplying two powers that have the same base, you simply add their exponents. Mathematically, it is represented as:

• \(a^m \times a^n = a^{m+n}\)

This rule helps in reducing expressions into fewer terms by combining exponents.

For example, if you have \(x^{12}\) and \(x^{-12}\), applying the product of powers rule means you calculate \(x^{12-12} = x^0\), which simplifies to 1, because any number raised to the 0 power is 1. This is precisely what was applied in the step-by-step solution, where terms with the same base were combined.

Understanding and applying the product of powers efficiently allows for a cleaner, simplified final expression.