Simplify square root of 5e^p
The given question asks to perform a mathematical operation on a given expression. The expression is made up of two mathematical constants: the square root and Euler's number ('e'). The square root function is typically represented with a radical sign, while 'e' represents Euler's number, which is an irrational and transcendental number approximately equal to 2.71828. The 'p' in the exponent position represents a variable whose value would affect the outcome of the expression. The task involves simplifying this expression by applying rules of exponents and square roots. Simplification usually entails rewriting the expression in a simpler or more standardized form without changing its value.
$\sqrt{5} e^{p}$
Answer
Solution:
Step 1:
To simplify $\sqrt{5e^p}$, we first need to recognize that the square root applies to both 5 and $e^p$ separately. This gives us:
$\sqrt{5} \cdot \sqrt{e^p}$
Step 2:
Next, we simplify the square root of the exponential. Since $e^p$ is already a simple exponential form, we only need to apply the square root to it:
$\sqrt{5} \cdot e^{\frac{p}{2}}$
Step 3:
Now, we combine the simplified square root and the exponential to get the final simplified expression:
$\sqrt{5} e^{\frac{p}{2}}$
Knowledge Notes:
To simplify an expression involving a square root of a product, such as $\sqrt{ab}$, we can use the property of square roots that states $\sqrt{ab} = \sqrt{a} \cdot \sqrt{b}$. This allows us to separate the square root of a product into the product of two separate square roots.
When dealing with the square root of an exponential, such as $\sqrt{e^p}$, we can use the property of exponents that states $e^{p/q} = \sqrt[q]{e^p}$. For a square root, which is the second root, this simplifies to $e^{p/2} = \sqrt{e^p}$.
It's important to note that these simplifications are valid for real numbers where the operations are defined. For complex numbers or other cases where the base of the exponential is negative, additional considerations may be necessary.
