Multiply 2 square root of 5( square root of 6+2)
The question asks you to perform a multiplication involving radical expressions, also known as square roots. You have a binomial inside one square root, consisting of the square root of 6 and the number 2, which needs to be multiplied by 2 times the square root of 5. The problem requires knowledge of the distributive property to multiply each term in the binomial by the term outside the parentheses.
$2 \sqrt{5} \left(\right. \sqrt{6} + 2 \left.\right)$
Answer
Solution:
Step 1:
Utilize the distributive property to expand the expression: $2\sqrt{5}(\sqrt{6} + 2)$.
Step 2:
Begin by multiplying $2\sqrt{5}$ with $\sqrt{6}$.
Step 2.1:
Apply the rule for multiplying radicals: $2\sqrt{5}\sqrt{6} = 2\sqrt{5 \times 6}$.
Step 2.2:
Calculate the product inside the radical: $2\sqrt{30}$.
Step 3:
Now, multiply $2\sqrt{5}$ by $2$: $2\sqrt{5} \times 2 = 4\sqrt{5}$.
Step 4:
Combine the results to express the solution in its various forms.
Exact Form: $2\sqrt{30} + 4\sqrt{5}$
Decimal Form: $19.89872306 \ldots$
Knowledge Notes:
The problem involves simplifying a mathematical expression that includes radicals (square roots) and the application of the distributive property. The distributive property states that for any numbers a, b, and c, the expression a(b + c) is equal to ab + ac.
To solve the problem, we followed these steps:
Distributive Property: This property is used to expand an algebraic expression of the form a(b + c). It is applied by multiplying the term outside the parentheses by each term inside the parentheses.
Product Rule for Radicals: When multiplying two radicals with the same index, the product rule states that $\sqrt{a} \times \sqrt{b} = \sqrt{a \times b}$.
Simplification: After applying the product rule, we simplify the expression by multiplying any numerical coefficients and evaluating any possible simplifications within the radicals.
Final Expression: The result is presented in two forms: the exact form, which maintains the radical notation, and the decimal form, which is a numerical approximation of the exact form.
In this problem, we used the distributive property to expand the given expression and then applied the product rule for radicals to simplify it. The final step was to multiply the remaining numerical coefficients to arrive at the simplified expression.
