Problem 42 - 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. $$ \left(\frac{1}{2} f^{2} r^{6} s^{5}\right)^{4} $$

Step-by-Step Solution

Answer

Answer: $\frac{1}{16} f^8 r^{24} s^{20}$

Step 1: Analyzing the expression and applying Power of a Power rule

Expression: $$ \left(\frac{1}{2} f^{2} r^{6} s^{5}\right)^{4} $$ This expression has the form \((a b^n c^m)^k\) where \(a=\frac{1}{2}\), \(b=f\), \(n=2\), \(c=r\), \(m=6\), and \(k=4\). We must apply the Power of a Power rule which states if you have \((a b^n c^m)^k\), you can distribute the exponent \(k\) to all factors in the expression. Thus, $$ \left(\frac{1}{2} f^{2} r^{6} s^{5}\right)^{4} = \left(\frac{1}{2}\right)^{4}\times \left(f^{2}\right)^{4}\times \left(r^{6}\right)^{4} \times \left(s^{5}\right)^{4} $$

Step 2: Applying Power of a Power rule to the exponents

We will now apply the power of a power rule to the exponents, which means \((b^n)^k = b^{n \times k}\). $$ \left(\frac{1}{2}\right)^{4}\times \left(f^{2}\right)^{4}\times \left(r^{6}\right)^{4}\times \left(s^{5}\right)^{4} = \left(\frac{1}{2}\right)^{4} \times f^{2\times4} \times r^{6\times4} \times s^{5\times4} $$

Step 3: Simplifying the expression

Now we must simplify the expression to get the final result. $$ \begin{aligned} \left(\frac{1}{2}\right)^{4} \times f^{2\times4} \times r^{6\times4} \times s^{5\times4} &= \left(\frac{1}{2}\right)^{4} \times f^{8} \times r^{24} \times s^{20} \\ &= \frac{1}{16} \times f^{8} \times r^{24} \times s^{20} \end{aligned} $$ The simplified expression is: $$ \frac{1}{16} f^8 r^{24} s^{20} $$

Key Concepts

Power Rules

Understanding power rules can be incredibly helpful when dealing with exponents. One of the most important power rules is the Power of a Power rule. This rule tells us how to handle situations where a power is raised to another power. For example, if you have an expression like \((b^n)^k\), the rule allows you to multiply the exponents: \(b^{n \times k}\).

In the problem you're solving, the whole expression \(\left(\frac{1}{2} f^{2} r^{6} s^{5}\right)^{4}\) utilizes this rule by distributing the exponent 4 to each part of the expression.

There's also the concept of the "Power of a Product" rule. This applies to expressions like \((abc)^n\), meaning that each factor in the product should be raised to the power. Applying it consistently leads to

• \((a)^n\)
• \((b)^n\)
• \((c)^n\)

Thus, expression like \((f^2 r^6 s^5)^4\) becomes \((f^2)^4\), \((r^6)^4\), and \((s^5)^4\).

Power rules are foundational, and understanding them makes simplifying expressions much simpler!

Simplifying Expressions

Simplifying expressions is all about making a mathematical expression easier to work with without changing its value. This process often involves using power rules, distributing exponents, and reducing complex numbers.

In the given exercise, after applying the power of a power rule, we obtain individual terms: \(\left(\frac{1}{2}\right)^{4} \times f^{8} \times r^{24} \times s^{20}\). Each term can be further simplified.

First, calculate \(\left(\frac{1}{2}\right)^4\). This involves taking 2 to the fourth power, which results in \(16\), and then inverting it, giving you \(\frac{1}{16}\). For the rest of the variables like \(f, r, \) and \(s\), the multiplication of exponents has already simplified them

• \(f^{2 \times 4} = f^8\)
• \(r^{6 \times 4} = r^{24}\)
• \(s^{5 \times 4} = s^{20}\)

Once the powers are computed, the expression is assembled into the simplest form possible, \(\frac{1}{16} f^8 r^{24} s^{20}\), making equations and expressions cleaner and easier to work with.

Variables

Variables are placeholders used to represent numbers in mathematical expressions and equations. They allow us to write general mathematical statements and laws that can apply in many situations.

In the expression \(\left(\frac{1}{2} f^{2} r^{6} s^{5}\right)^{4}\), the variables \(f\), \(r\), and \(s\) stand in for values that could be any number. This abstraction is what makes algebra so powerful!

When you raise a variable to a power, as in \(f^2\), you're indicating repeated multiplication: \(f \times f\). Understanding this concept is key when you're manipulating algebraic expressions.

Variables remain in the expression even after the power rules are applied, forming part of the simplified expression. In this instance, the application of power rules alters only their exponents, not the variables themselves. Thus, they are carried through each simplification step, ending in the final expression \(\frac{1}{16} f^8 r^{24} s^{20}\).

Figuring out how variables interact with each other and with constants in an equation is an essential skill in mathematics and sciences, helping to solve real-world problems.