Problem 55 - Elementary Algebra

Question

Use the power rules for exponents to simplify the following problems. Assume that all bases are nonzero and that all variable exponents are natural numbers. $$ \frac{\left(m^{5} n^{6} p^{4}\right)^{4}}{\left(m^{4} n^{5} p\right)^{4}} $$

Step-by-Step Solution

Answer

Question: Simplify the expression \(\frac{(m^5n^6p^4)^4}{(m^4n^5p)^4}\). Answer: The simplified expression is \(m^4n^4p^{12}\).

Step 1: Apply The Power of a Power Rule

First, raise each term inside the parentheses to the fourth power: $$ \frac{(m^{5*4}n^{6*4}p^{4*4})}{(m^{4*4}n^{5*4}p^{1*4})} $$

Step 2: Calculate The Powers

Now multiply each exponent by four: $$ \frac{m^{20}n^{24}p^{16}}{m^{16}n^{20}p^4} $$

Step 3: Apply The Quotient of Powers Rule

Divide each term with the same base to simplify the expression further: $$ m^{20-16}n^{24-20}p^{16-4} $$

Step 4: Calculate The Differences

Now subtract the exponents to get the simplified expression: $$ m^{4}n^{4}p^{12} $$ The simplified expression is \(m^4n^4p^{12}\).

Key Concepts

Power of a Power Rule

When you come across an expression that involves an exponent raised to another exponent, it's time to use the power of a power rule. This rule is handy for simplifying expressions where each base is multiplied by the same number. If you have a base raised to one exponent, and then that whole setup is raised again to another exponent, it goes like this: you multiply those exponents together.

For example, given \((a^m)^n\), you transform it to \(a^{m \times n}\).

This is just what we do in the first step of our example. We took each term inside the parentheses \( (m^5 n^6 p^4)^4 \) and used the power of a power rule to multiply the exponents by 4. This step ensures we simplify the expression correctly right from the start. Remember, every base gets its exponent multiplied independently by the outer exponent.

Quotient of Powers Rule

In situations where you have powers with the same base being divided, the quotient of powers rule is your friend for simplifying. This rule tells us that you subtract the exponent in the denominator from the exponent in the numerator. It's an efficient way to break down expressions and solve them step by step.

Let's consider \( \frac{a^m}{a^n} = a^{m-n} \). Here, you'd subtract \(n\) from \(m\) to get your new exponent.

Applying this rule to our example, once we've raised each base to the necessary power, our expression was \((m^{20} n^{24} p^{16})/(m^{16} n^{20} p^4)\). By using the quotient of powers rule, we simplified it further by performing the subtraction for each corresponding base's exponent. This rule helps to streamline your calculations, offering a clear path toward the simplest form of an expression.

Simplifying Expressions

Simplifying expressions is all about breaking down and rewriting expressions in their easiest, most concise form. After applying rules like the power of a power and quotient of powers, you're often almost there. The goal of simplification is to have a cleaner, more efficient way of representing an expression.

This involves rewriting any terms where arithmetic can be performed, dealing with like bases, and ensuring that all possible operations are completed.

• Identify and apply necessary rules.
• Ensure all calculations for exponents are done properly.
• Double-check that all terms have been simplified.

In our exercise, we followed through from ordering our tasks, applying the exponent rules, and then simplifying step by step. From getting \(m^4 n^4 p^{12}\) out of a complex fraction, each operation was necessary for achieving the simplest form. Always aim for accuracy, as each mathematical adjustment has a reason behind it.