Problem 251 - Elementary Algebra

Question

In the following exercises, simplify. $$ \left(5 \sqrt{2 d^{7}}\right)\left(3 \sqrt{50 d^{3}}\right) $$

Step-by-Step Solution

Answer

\(150d^{5}\)

Step 1 - Multiply the coefficients

First, multiply the coefficients 5 and 3. This gives: \[ 5 \times 3 = 15 \]

Step 2 - Multiply the square roots

Inside the square roots, multiply the terms together: \[ \sqrt{2d^{7}} \times \sqrt{50d^{3}} = \sqrt{(2d^{7})(50d^{3})} \]

Step 3 - Simplify inside the square root

Simplify the expression inside the square root by multiplying the constants and combining the exponents for the variable d: \[ 2 \times 50 = 100 \] and \[ d^{7} \times d^{3} = d^{10} \] This results in: \[ \sqrt{100d^{10}} \]

Step 4 - Simplify the square root

Take the square root of the resulting expression: \[ \sqrt{100d^{10}} = 10d^{5} \]

Step 5 - Multiply the simplified result by the coefficient

Finally, multiply the simplified result by the coefficient from Step 1: \[ 15 \times 10d^{5} = 150d^{5} \]

Key Concepts

Multiplying Coefficients

To simplify algebraic expressions involving multiplication, start by handling the coefficients. Coefficients are the numerical parts of terms. In the given exercise, we have the coefficients 5 and 3 in the terms \(5 \sqrt{2 d^{7}}\) and \(3 \sqrt{50 d^{3}}\). Multiplying these coefficients is straightforward:
\. The result is a simpler coefficient that will be used in the next steps. This method lays the groundwork for making the entire expression more manageable.

Multiplying Radicals

Multiplying radicals involves combining the radical expressions under a single square root. In our exercise, we have: \(\sqrt{2d^7} \times \sqrt{50d^3}\). When multiplying these, you combine the terms inside the radicals:
\[ \sqrt{(2d^7) \times (50d^3)} \] On simplifying inside the radicals, you get:
\[ \sqrt{100d^{10}}. \] This combined radical now needs simplifying, making the expression more compact.

Combining Exponents

Combining exponents is essential for managing terms with variables. When you multiply expressions with the same base, you add the exponents:
\[ d^7 \times d^3 \] results in
\[ d^{7 + 3} = d^{10}\]. When simplified within our radical form, this becomes part of a single expression under the square root. Properly combining exponents ensures you have fewer, more simplified terms to work with.

Taking Square Roots

Taking square roots is the final key step in our process. After multiplying and simplifying under the radical, you end with
\[ \sqrt{100d^{10}}\]. To take the square root, recognize that \[ \sqrt{100} = 10\] and \[ \sqrt{d^{10}} = d^{5}\]. Thus, the square root of the entire expression is:
\[ 10d^5\]. Don't forget to multiply this result by the coefficient from the multiplication of initial coefficients, giving you:
\[ 15 \times 10d^5 = 150d^5. \] Taking square roots effectively simplifies expressions to their most reduced form.