Problem 67 - Algebra for College Students

Question

Simplify each of the following. Express final results using positive exponents only. For example,\(\left(2 x^{\frac{1}{2}}\right)\left(3 x^{\frac{1}{3}}\right)=6 x^{\frac{5}{6}}\). \(\left(8 x^{6} y^{3}\right)^{\frac{1}{3}}\)

Step-by-Step Solution

Answer

The expression simplifies to \(2x^2y\).

Step 1: Apply the Exponent Rule

The term \((8x^6y^3)^\frac{1}{3}\) involves raising a product to a fractional power. Use the rule \((a \cdot b)^n = a^n \cdot b^n\). This means we should distribute the \(\frac{1}{3}\) exponent to each part of the product inside the parentheses.

Step 2: Simplify Each Component

Apply the exponent individually to each part of the expression:1. \((8)^\frac{1}{3}\): Calculate the cube root of 8, which is 2, because \(2^3 = 8\).2. \((x^6)^\frac{1}{3}\): Use the rule \((a^m)^n = a^{m \cdot n}\). Thus, \(x^{6 \cdot \frac{1}{3}} = x^2\).3. \((y^3)^\frac{1}{3}\): Using the same rule, \(y^{3 \cdot \frac{1}{3}} = y^1 = y\).

Step 3: Combine Simplified Components

Now combine the results from each component:The expression simplifies to \(2 \cdot x^2 \cdot y\). Thus, the simplified form of \((8x^6y^3)^\frac{1}{3}\) is \(2x^2y\).

Key Concepts

Exponent Rules

Exponent rules are foundational tools in algebra that help us manipulate expressions involving powers. When we talk about exponents, we're referring to repeated multiplication. For example, in an expression like \(x^a\), \(x\) is the base and \(a\) is the exponent, indicating \(x\) is multiplied by itself \(a\) times.

Whenever we encounter expressions with multiple bases raised to a power, we use the product rule for exponents. This rule states that \((a \cdot b)^n = a^n \cdot b^n\), meaning you can distribute the power to each component in a product. This is particularly useful when dealing with complex expressions.

Additionally, there's the power of a power rule. For any base \(a\) raised to a power and then raised again to another power, \((a^m)^n = a^{m \cdot n}\). This helps us simplify expressions like \((x^6)^{1/3}\), where we multiply the exponents to get \(x^2\). These essential rules make manipulating algebraic expressions much more straightforward.

Simplifying Expressions

Simplifying expressions is all about making them as easy to work with as possible. The goal is to rewrite expressions in their simplest form while following mathematical rules.

To simplify an expression involving exponents, apply the exponent rules carefully to distribute powers across the numbers and variables within the expression. For example, with \((8x^6y^3)^{1/3}\), we use the rule \((a \cdot b)^n = a^n \cdot b^n\) to distribute \(1/3\) to each part of the expression.

While simplifying, evaluate numerical bases directly. For instance, the cube root of 8 is simply 2. For variables, adjust the exponents according to exponent multiplication rules, like changing \(x^{6 \cdot \frac{1}{3}}\) to \(x^2\). Always ensure all simplified components are recombined to reach the final expression.

Positive Exponents

When working with expressions, it's often required to express all terms with positive exponents. Positive exponents are easier to interpret and handle compared to negative or fractional exponents.

Positive exponents indicate how many times a number should be multiplied by itself. When simplifying, take care to convert all exponents to positive values where possible. In our example, \(y^{3 \cdot \frac{1}{3}}\) simplifies directly to \(y\), ensuring we maintain a positive exponent.

If you encounter a negative exponent, remember that \(a^{-n} = \frac{1}{a^n}\). Using these principles, arranging components to end up with positive exponents ensures clarity and simplicity in algebraic expressions. This approach will not only make further calculations easier but will also help in understanding how terms relate to each other in an expression.