Problem 43 - 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. $$ \left(\frac{1}{8} c^{10} d^{8} e^{4} f^{9}\right)^{2} $$

Step-by-Step Solution

Answer

Question: Simplify the given expression $$\left(\frac{1}{8} c^{10} d^{8} e^{4} f^{9}\right)^{2}$$ assuming all variable exponents are natural numbers. Answer: $$\frac{1}{64} c^{20} d^{16} e^{8} f^{18}$$

Step 1: Apply the power rule to the fraction

Raise the entire fraction inside the power to the power of 2:$$ \left(\frac{1}{8} c^{10} d^{8} e^{4} f^{9}\right)^{2} = \frac{1^2}{8^2} c^{10 \cdot 2} d^{8 \cdot 2} e^{4 \cdot 2} f^{9 \cdot 2} $$

Step 2: Simplify the rational expression

Simplify the fraction by squaring the numerator and denominator:$$ \frac{1^2}{8^2} = \frac{1}{64} $$

Step 3: Apply the power rule to the variable exponents

Multiply the exponents of each variable by 2:$$ c^{10 \cdot 2} = c^{20} \\ d^{8 \cdot 2} = d^{16} \\ e^{4 \cdot 2} = e^{8} \\ f^{9 \cdot 2} = f^{18} $$

Step 4: Write the simplified expression

Combine the results from Steps 2 and 3 to write the simplified expression:$$ \frac{1}{64} c^{20} d^{16} e^{8} f^{18} $$The simplified expression is:$$ \frac{1}{64} c^{20} d^{16} e^{8} f^{18} $$

Key Concepts

Understanding Power Rules

The power rules are essential when working with exponents and simplifying expressions. When we have a base raised to a power and then raised to another power, we multiply the exponents. This is known as the power of a power rule. For instance, \( (a^m)^n = a^{m imes n} \). This might seem tricky at first, but think of it as repeated multiplication. Each set of parentheses means you are multiplying the base by itself the number of times specified by the product of the exponents.
For example:

• If you begin with \( c^{10} \) and square it, you will multiply 10 by 2, resulting in \( c^{20} \).
• The same rule applies to any other base, whether it looks complicated like \( \left(\frac{1}{8} c^{10} d^{8} e^{4} f^{9}\right) \) or simple like \( x^3 \).

The power rules simplify expressions by shrinking the number of operations you need to think about.

Breaking Down Exponents

Exponents indicate how many times a number, called the base, is multiplied by itself. Understanding how to manipulate exponents is crucial in algebra. When we see a term like \( x^n \), it means we multiply \( x \) by itself \( n \) times.
There are some important rules to remember with exponents:

• If you multiply powers with the same base, you add the exponents: \( a^m \times a^n = a^{m+n} \).
• If you divide powers with the same base, you subtract the exponents: \( \frac{a^m}{a^n} = a^{m-n} \).
• Raising a fraction to a power means raising both the numerator and denominator to that power: \( \left(\frac{a}{b}\right)^n = \frac{a^n}{b^n} \).

These rules allow us to manipulate and simplify expressions that include exponents efficiently.

Simplifying Expressions

Simplifying expressions involves reducing an expression to its simplest form while maintaining its value. This often requires applying the power rules for exponents effectively. In our exercise, we simplified the expression \( \left(\frac{1}{8} c^{10} d^{8} e^{4} f^{9}\right)^2 \) by first applying the power rule to each part of the fraction and the variables.
To simplify:

• First, handle any fractions by raising both the numerator and denominator to the given power. In this case, we have \( \frac{1}{8} \). Squaring yields \( \frac{1^2}{8^2} = \frac{1}{64} \).
• Next, apply the power of a power rule to each variable: multiply the exponent by 2. For example, \( c^{10} \) becomes \( c^{20} \).

After applying these steps, the expression is simplified to \( \frac{1}{64} c^{20} d^{16} e^{8} f^{18} \), making it easier to work with or evaluate further.