Simplify square root of 3(7-5 square root of 5)
The question is asking for the simplification of a mathematical expression that involves square roots and basic arithmetic operations. Specifically, you are required to evaluate and simplify the product of the square root of 3 and an expression in parentheses, which consists of the difference between the number 7 and the product of 5 and the square root of 5. The objective is to apply the rules of arithmetic and simplification to represent the expression in a more elementary or reduced form.
$\sqrt{3} \left(\right. 7 - 5 \sqrt{5} \left.\right)$
Answer
Solution:
Step 1:
Utilize the distributive law to expand $\sqrt{3} \times (7 - 5\sqrt{5})$.
Step 2:
Reposition $7$ before $\sqrt{3}$ to get $7 \times \sqrt{3} + \sqrt{3} \times (-5\sqrt{5})$.
Step 3:
Perform the multiplication of $\sqrt{3} \times (-5\sqrt{5})$.
Step 3.1:
Apply the rule for multiplying square roots to get $7 \times \sqrt{3} - 5\sqrt{3 \times 5}$.
Step 3.2:
Calculate the product of $3$ and $5$ to obtain $7 \times \sqrt{3} - 5\sqrt{15}$, which simplifies to $7\sqrt{3} - 5\sqrt{15}$.
Step 4:
Present the final expression in its various forms:
- Exact Form: $7\sqrt{3} - 5\sqrt{15}$
- Decimal Form: Approximately $-7.24056107 \ldots$
Knowledge Notes:
The problem involves simplifying a radical expression using algebraic properties. Here are the relevant knowledge points:
Distributive Property: This property allows us to multiply a single term by each term within a parenthesis. For example, $a(b + c) = ab + ac$.
Multiplication of Radicals: When multiplying radicals, we can use the product rule, which states that $\sqrt{a} \times \sqrt{b} = \sqrt{a \times b}$, provided that $a$ and $b$ are non-negative.
Combining Like Terms: Terms that have the same radical part can be combined by adding or subtracting their coefficients.
Simplifying Radicals: When a radical contains a perfect square, it can be simplified by taking the square root of that perfect square. For example, $\sqrt{16} = 4$.
Decimal Approximation: Sometimes, it is useful to represent an exact expression, especially one involving radicals, as a decimal approximation for practical purposes. This is often done using a calculator or computer software.
In this problem, the distributive property is first applied, followed by the multiplication of radicals and simplification of the resulting expression. The final answer is provided in both exact and decimal forms.
