- #1
Albert1
- 1,221
- 0
simplify:
$\sqrt {21-4 \sqrt 5 +8\sqrt 3 - 4\sqrt {15}}$
$\sqrt {21-4 \sqrt 5 +8\sqrt 3 - 4\sqrt {15}}$
MarkFL said:Here is my solution:
\(\displaystyle 21+8\sqrt{3}-4\sqrt{5}-4\sqrt{15}=\)
\(\displaystyle 4+4\sqrt{3}-2\sqrt{5}+4\sqrt{3}+12-2\sqrt{15}-2\sqrt{5}-2\sqrt{15}+5=\)
\(\displaystyle \left(2+2\sqrt{3}-\sqrt{5} \right)^2\)
Hence:
\(\displaystyle \sqrt{21+8\sqrt{3}-4\sqrt{5}-4\sqrt{15}}=2+2\sqrt{3}-\sqrt{5}\)
A radical expression is an expression that contains a radical, or square root, symbol. It is a mathematical representation of a number or quantity that cannot be simplified into a whole number or fraction.
To simplify a radical expression, you must find the factors of the number inside the radical and rewrite it using perfect squares. Then, you can take the square root of the perfect squares to simplify the expression further.
Yes, you can simplify a radical expression with variables by following the same steps as simplifying a radical expression with numbers. However, if the variables have exponents, you must apply the exponent rules before simplifying.
A radical expression contains a radical symbol, while a rational expression contains a fraction with variables in the numerator and denominator. Simplifying a radical expression involves finding perfect squares, while simplifying a rational expression involves finding common factors.
Simplifying radical expressions can be useful in various mathematical and scientific contexts. It can help in solving equations, simplifying complicated expressions, and simplifying measurements and units in physics and engineering problems.