Problem 571 - Elementary Algebra

Question

In the following exercises, simplify. (a) \(\left(a^{12}\right)^{\frac{1}{6}}\) (b) \(\left(b^{15}\right)^{\frac{3}{5}}\) (c) \(\left(c^{11}\right)^{\frac{1}{11}}\)

Step-by-Step Solution

Answer

(a) a^2 (b) b^9 (c) c

Step 1: Understand the Power Rule for Exponents

The power rule for exponents states that \(\left(a^m\right)^n = a^{(m*n)}\). This rule will help in simplifying the given expressions.

Step 2: Simplify Part (a)

Given \(\left(a^{12}\right)^{\frac{1}{6}}\), apply the power rule: \(\left(a^{12}\right)^{\frac{1}{6}} = a^{12*\frac{1}{6}} = a^2\).

Step 3: Simplify Part (b)

Given \(\left(b^{15}\right)^{\frac{3}{5}}\), apply the power rule: \(\left(b^{15}\right)^{\frac{3}{5}} = b^{15*\frac{3}{5}} = b^9\).

Step 4: Simplify Part (c)

Given \(\left(c^{11}\right)^{\frac{1}{11}}\), apply the power rule: \(\left(c^{11}\right)^{\frac{1}{11}} = c^{11*\frac{1}{11}} = c\).

Key Concepts

Power Rule

The power rule is a fundamental concept when dealing with exponents. It states that when you have a power raised to another power, you simply multiply the exponents. Formally, the rule is given by \[\big(a^m\big)^n = a^{m\cdot n}\]. This rule makes it easier to handle complex expressions and simplifies the overall problem-solving process. For example, in the problem \[\big(a^{12}\big)^{1/6}\], using the power rule, you multiply the exponents 12 and 1/6 to get \[\big(a^{12\cdot1/6}\big) = a^2\]. Understanding this rule can significantly streamline your approach to handling exponential expressions.

Algebraic Simplification

Algebraic simplification involves reducing expressions to their simplest form. This process often includes combining like terms, factoring polynomials, and applying rules like the power rule. The goal is to make the expression as simple as possible for easier understanding and solving.

In the given problems, simplifying \[\big(b^{15}\big)^{3/5}\] involves using the power rule to yield \[\big(b^{15\cdot3/5}\big) = b^9\]. Here, multiplying the exponents simplifies the expression, making it more straightforward to understand or solve any further calculations involving this term.

Exponent Laws

Exponent laws are the rules that govern how exponents can be manipulated in mathematical expressions. These laws include but are not limited to the power rule, the product rule \[\big(a^m \cdot a^n = a^{m+n}\big)\], and the quotient rule \[\big(a^m / a^n = a^{m-n}\big)\].

For instance, in \[\big(c^{11}\big)^{1/11}\], you apply the power rule to get \[\big(c^{11\cdot1/11}\big) = c\]. This simplification is crucial as it turns a complicated expression into a basic one, which is much easier to manage and solve. Mastering these laws is essential for anyone tackling algebraic problems involving exponents.