Problem 120 - Intermediate Algebra

Question

Simplify. See a Concept Check in this section. Assume variables represent positive numbers. $$ \sqrt[3]{a^{9} b^{21} c^{3}} $$

Step-by-Step Solution

Answer

The expression simplifies to \(a^3 b^7 c\).

Step 1: Identify the Expression

The expression given is \( \sqrt[3]{a^{9} b^{21} c^{3}} \). This is a cube root involving the variables \(a\), \(b\), and \(c\), each raised to different powers.

Step 2: Rewrite Expression with Fractional Exponents

Rewrite the entire expression using fractional exponents. The cube root of \(a^{9}b^{21}c^{3}\) can be rewritten as \((a^{9}b^{21}c^{3})^{1/3}\).

Step 3: Apply Exponent Rule

Apply the rule \((x^m)^n = x^{m \cdot n}\) to the expression. This allows us to distribute the fractional exponent \(1/3\) to each term inside the parentheses: \(a^{9 \cdot (1/3)} b^{21 \cdot (1/3)} c^{3 \cdot (1/3)}\).

Step 4: Simplify Exponents

Calculate each of the exponent multiplications: - \(9 \times \frac{1}{3} = 3\), - \(21 \times \frac{1}{3} = 7\), - \(3 \times \frac{1}{3} = 1\). Therefore, the expression simplifies to \(a^3 b^7 c^1\).

Step 5: Express the Final Simplified Expression

Combine the simplified terms: The expression \(a^3 b^7 c^1\) can be written as \(a^3 b^7 c\) since any variable raised to the power of 1 is just the variable itself.

Key Concepts

Fractional Exponents

Fractional exponents might seem tricky at first, but they are just another way to express roots. Here’s a simple breakdown: an exponent that is a fraction, such as \( \frac{1}{3} \), represents a root. In this case, \( x^{\frac{1}{3}} \) is the cube root of \( x \). Similarly, \( x^{\frac{1}{2}} \) would be the square root of \( x \).

When converting cube roots into fractional exponents, like in the expression \( \sqrt[3]{a^{9}b^{21}c^{3}} \), you can rewrite it as \( (a^{9}b^{21}c^{3})^{1/3} \). Fractions in exponents allow us to handle roots using multiplication and make it easier to apply other exponent rules later on.

Using fractional exponents is a convenient way to simplify complex expressions, especially when multiple variables and roots are involved.

Exponent Rules

Exponent rules are wonderful tools that make working with powers much easier. A key rule to know is \((x^m)^n = x^{m \cdot n}\). This means that when you raise an exponent to another exponent, you can multiply the exponents together.

In our exercise, we used this rule by applying the fractional exponent \(1/3\) to each individual term:

• \(a^{9 \cdot \frac{1}{3}} = a^3\)
• \(b^{21 \cdot \frac{1}{3}} = b^7\)
• \(c^{3 \cdot \frac{1}{3}} = c^1\)

This process simplifies the expression significantly. Once you have applied this rule, simplifying further becomes much easier. Remember, understanding these rules helps you tackle even the most challenging equations with confidence!

Simplifying Expressions

Simplifying expressions is all about making complex mathematical statements easier to work with. After you’ve applied your exponent rules, the goal is to break down the expression into its simplest form.

In the given exercise, after applying the fractional exponent and exponent rule, we simplified \(a^3 b^7 c^1\) to \(a^3 b^7 c\). The rule here is that any number or variable to the power of 1 is simply the number or variable itself.

Why simplify? Because a simpler expression is easier to understand and use in further calculations. Simplification is about finding the cleanest, most straightforward version of a formula or expression, making it more intuitive to work with in algebra and beyond.

By practicing simplification, you develop a keen eye for recognizing patterns and shortcuts in math, significantly aiding problem-solving skills.