Problem 38 - 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. $$ \left(m^{6} n^{2} p^{5}\right)^{5} $$

Step-by-Step Solution

Answer

Question: Simplify the expression \(\left(m^{6} n^{2} p^{5}\right)^{5}\). Answer: \(m^{30} n^{10} p^{25}\)

Step 1: Identify the expression and exponents

The given expression is \(\left(m^{6} n^{2} p^{5}\right)^{5}\), with the variables \(m\), \(n\), and \(p\) having exponents of 6, 2, and 5 respectively. The entire expression is raised to the power of 5.

Step 2: Apply the Power of a Power rule

To apply the Power of a Power rule, we must multiply the exponent of each of the variables by the exponent of the outer term (5). So for each variable, we will perform the following calculations: 1. \(m^6\): Multiply 6 by the outer exponent 5: \(6 * 5 = 30.\) 2. \(n^2\): Multiply 2 by the outer exponent 5: \(2 * 5 = 10.\) 3. \(p^5\): Multiply 5 by the outer exponent 5: \(5 * 5 = 25.\)

Step 3: Rewrite the expression with the new exponents

Using the results from step 2, we rewrite the expression with the new exponents: $$ \left(m^{6} n^{2} p^{5}\right)^{5} = m^{30} n^{10} p^{25} $$

Step 4: Simplify the expression

The expression is already simplified. The final simplified form is: $$ m^{30} n^{10} p^{25} $$

Key Concepts

Simplifying Expressions with Exponents

When you encounter expressions with exponents, simplifying them makes these expressions much more manageable and easier to understand. Simplifying means reducing the expression to its simplest form without changing its value.

To simplify expressions with exponents, you first need to know the specific rules that govern exponent operations. These rules help you figure out how to handle expressions that involve multiplication, division, or raising powers to other powers. In the given exercise, we focus on the multiplication of exponents during the process of simplifying.

For instance, when you have multiple exponents with the same base being multiplied, you can add the exponents. If you have \(a^{m} \cdot a^{n}\), you can simplify it as \(a^{m+n}\). However, when you have a power to a power, such as our problem \(\left(m^{6} n^{2} p^{5}\right)^{5}\), you multiply the exponents, which we will delve into more in the next section.

Power of a Power Rule

The power of a power rule is a fundamental principle in algebra that helps simplify expressions where an exponent is raised to another exponent. The rule states that when you have \(\left(a^{n}\right)^{m}\), you can simplify it as \(a^{n \cdot m}\). This means you multiply the exponents to find the new exponent of the base number.

In the exercise, you have the expression \(\left(m^{6} \cdot n^{2} \cdot p^{5}\right)^{5}\). Applying the power of a power rule, you end up multiplying each individual exponent by the outer exponent of 5, simplifying it to \(m^{30} \cdot n^{10} \cdot p^{25}\). This condensed form of the original expression is much simpler to work with, whether you are continuing with algebraic operations or substituting numerical values for the variables.

Exponent Multiplication

Multiplying exponents comes into play when you are dealing with the power of a power rule or when multiplying bases with exponents. When bases are different and exponents are similar, we cannot add or multiply the exponents directly, but in situations where the same base is being raised to a power that is then raised to another power, exponent multiplication becomes essential.

In our original exercise, \(m^{6} \) is raised to the power of 5, as is the case with \(n^{2} \) and \(p^{5}\). What we're essentially doing is taking the exponent on the base (6, 2, and 5 respectively) and multiplying it by the outer exponent (5). This allows us to maintain the integrity of the expression's value while simplifying its form. It's important to remember that this operation is only valid when the bases remain the same and is not applicable to different bases being multiplied together.