Problem 63 - 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. $$ \left(\frac{3 a^{2} b^{3}}{c^{4}}\right)^{3} $$

Step-by-Step Solution

Answer

Question: Simplify the expression $\left(\frac{3 a^{2} b^{3}}{c^{4}}\right)^{3}$ using the power rules for exponents. Answer: $\frac{27 a^6 b^9}{c^{12}}$

Step 1: Apply the Power Rule to Each Term

To simplify the expression, we will start by applying the power rule to the whole expression: $$\left(\frac{3 a^{2} b^{3}}{c^{4}}\right)^{3}$$ Remember, the power rule states that for all nonzero bases a, b, and c, and positive integers m and n: $$(a^m)^n = a^{mn}$$ Applying this rule to each term inside the parentheses, we get: $$\left(\frac{3^3 (a^2)^3 (b^3)^3}{(c^4)^3}\right)$$

Step 2: Simplify Each Term

Now, we can simplify each term inside the parentheses with the resulting expressions: $$\frac{3^3 (a^{2\cdot3}) (b^{3\cdot3})}{(c^{4\cdot3})}$$ Now, evaluate the new exponents and the numerical term to get: $$\frac{27 a^6 b^9}{c^{12}}$$

Step 3: Write the Final Simplified Expression

Finally, we can write our simplified expression as the result: $$\frac{27 a^6 b^9}{c^{12}}$$ This is the simplified expression of the given problem using the power rules for exponents.

Key Concepts

Power Rules for Exponents

Understanding the power rules for exponents is essential for dealing with algebraic expressions that involve exponentiation. When an exponent is raised to another power, the 'power to a power' rule comes into play. This rule states that you can multiply the exponents together to simplify the expression.

For example, let's consider the expression \( (a^m)^n \). According to the power rule, we simplify this by multiplying the exponents: \( (a^m)^n = a^{mn} \). This process was applied in our exercise to simplify the more complex algebraic expression with ease.

When the base is a product, like \( (ab)^n \), each part of the product is raised to the power separately: \( (ab)^n = a^n b^n \). Similarly, when the base is a fraction \( \left(\frac{a}{b}\right)^n \), both the numerator and the denominator are raised to the power: \( \left(\frac{a}{b}\right)^n = \frac{a^n}{b^n} \). These rules are particularly helpful in simplifying terms and making complex expressions more manageable.

Exponentiation

Exponentiation is a mathematical operation involving two numbers, the base and the exponent. When we write \( a^n \), \( a \) is the base and \( n \) is the exponent, which tells us how many times to multiply the base by itself.

It's important to note that the base can be any real number, and the exponent indicates repeated multiplication. For natural number exponents, the definition is quite straightforward: \( a^3 = a \times a \times a \), but exponentiation can also work with negative exponents or fractional exponents, each following specific rules.

For students to grasp this concept firmly, it is helpful to practice with different bases and exponents to see how altering these values affects the outcome. This understanding will build a solid foundation for more advanced algebraic manipulations. Remember, exponentiation is governed by several properties and rules which make calculations more efficient and less error-prone.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and at least one arithmetic operation (addition, subtraction, multiplication, division, and exponentiation). In our exercise, the expression \( \left(\frac{3a^{2}b^{3}}{c^{4}}\right)^{3} \) is an example of an algebraic expression involving a fraction with variables in both the numerator and the denominator raised to a power.

The beauty of algebraic expressions lies in their use to represent real-world situations and the ability to manipulate them to solve problems. Simplifying these expressions is a skill that requires an understanding of the distributive property, combining like terms, and, as seen in this exercise, the power rules for exponents.

We use algebraic expressions to model relationships and changes, providing a tool to explore patterns and make predictions. Mastery of simplifying such expressions enables learners to tackle more complex equations and deepen their understanding of mathematical relationships.