Problem 47 - Elementary Algebra

Question

Simplify. $$ \sqrt{9 c^{8} d^{12}} $$

Step-by-Step Solution

Answer

The simplified form is \(3c^4d^6\).

Step 1 - Factor inside the square root

Identify and factor the expression inside the square root as much as possible: the expression \(\sqrt{9 c^{8} d^{12}}\)can be written as \(\sqrt{(3^2) (c^8) (d^{12})}\).

Step 2 - Apply the square root to each factor

The square root of a product is the product of the square roots: the expression can be separated as \(\sqrt{3^2} \times \sqrt{c^8} \times \sqrt{d^{12}}\).

Step 3 - Simplify each square root

Apply the square root to each term: the term \(\sqrt{3^2} = 3\), the term \(\sqrt{c^8} = c^{8/2} = c^4\), and the term \(\sqrt{d^{12}} = d^{12/2} = d^6\).

Step 4 - Combine the simplified terms

Now combine all the simplified terms: the expression becomes \(3c^4d^6\).

Key Concepts

Square Roots

Square roots are a fundamental concept in algebra. The square root of a number is a value that, when multiplied by itself, gives the original number. For instance, the square root of 9 is 3, because 3 multiplied by 3 equals 9. Visually, the square root is represented by the radical symbol \( \sqrt{} \). When dealing with square roots of expressions involving variables, we can apply the property that \( \sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b} \). This allows us to break down complex square root expressions into simpler parts.

Factoring

Factoring is the process of breaking down an expression into simpler components that, when multiplied together, give the original expression. In our example, the expression \( 9c^8d^{12} \) can be factored into \( (3^2)(c^8)(d^{12}) \). Factoring is particularly useful when simplifying radical expressions, as it allows us to more easily apply the square root to each component. Factoring makes it clear which parts of an expression can be simplified further under the square root.

Exponents

Exponents are a way to represent repeated multiplication of a number by itself. For example, \( c^8 \) means \( c \) multiplied by itself 8 times. A crucial rule to remember with exponents is \( (a^m)^n = a^{m \cdot n} \). When simplifying square roots, we can use the fractional exponent rule: \( \sqrt{a^b} = a^{b/2} \). In our problem, \( \sqrt{c^8} = c^{8/2} = c^4 \), and similarly, \( \sqrt{d^{12}} = d^{12/2} = d^6 \).

Algebraic Simplification

Algebraic simplification involves reducing an expression to its simplest form. For the given problem, algebraic simplification includes applying the square root to each factored term and then combining the results. Starting with \( \sqrt{9c^8d^{12}} \), we break it down factor by factor: \( \sqrt{3^2}, \sqrt{c^8}, \sqrt{d^{12}} \). Simplifying each: \( \sqrt{3^2} = 3, \sqrt{c^8} = c^4, \sqrt{d^{12}} = d^6 \). Combining these results, we get the final simplified form of \( 3c^4d^6 \), an elegant and reduced expression of the original problem.