Question
Simplify the radical expression. 5x7–√−8x7–√ (1 point) Responses −3x7–√ negative 3 x square root of 7 −21x negative 21 x −3x14−−√ negative 3 x square root of 14 13x7–√
Answers
GPT-4o mini
To simplify the expression \( 5 \sqrt{7} - \sqrt{-8} \sqrt{7} \), we first handle the term with the square root of a negative number.
Recall that \( \sqrt{-8} = \sqrt{8} \cdot \sqrt{-1} = 2\sqrt{2}i \). Therefore, we can rewrite the expression as follows:
\[
5 \sqrt{7} - \sqrt{-8} \sqrt{7} = 5 \sqrt{7} - 2\sqrt{2} i \sqrt{7}
\]
We can factor out \( \sqrt{7} \):
\[
\sqrt{7} \left( 5 - 2\sqrt{2} i \right)
\]
However, since the radical calculation involves imaginary numbers, this expression simplifies to a complex number.
If we only consider real numbers in this context and substitute \(\sqrt{-8}\) out of consideration, we get:
Given the options (-3x√7, -21x, -3x√14, 13x√7), it seems the expression involves merely real coefficients and radicals.
By examining the choices, the corresponding real interpretation or alternate formulations of complex expressions might yield -21x or similar choices that simplify within real confines (if other terms vanish or equate). Yet, from expectations of radical and complex interactions reformulating to other pure terms also could delve into roots.
Please provide the context around usage or limits set (or does employ real only) to ascertain the right applicable choice from provided formats.
Recall that \( \sqrt{-8} = \sqrt{8} \cdot \sqrt{-1} = 2\sqrt{2}i \). Therefore, we can rewrite the expression as follows:
\[
5 \sqrt{7} - \sqrt{-8} \sqrt{7} = 5 \sqrt{7} - 2\sqrt{2} i \sqrt{7}
\]
We can factor out \( \sqrt{7} \):
\[
\sqrt{7} \left( 5 - 2\sqrt{2} i \right)
\]
However, since the radical calculation involves imaginary numbers, this expression simplifies to a complex number.
If we only consider real numbers in this context and substitute \(\sqrt{-8}\) out of consideration, we get:
Given the options (-3x√7, -21x, -3x√14, 13x√7), it seems the expression involves merely real coefficients and radicals.
By examining the choices, the corresponding real interpretation or alternate formulations of complex expressions might yield -21x or similar choices that simplify within real confines (if other terms vanish or equate). Yet, from expectations of radical and complex interactions reformulating to other pure terms also could delve into roots.
Please provide the context around usage or limits set (or does employ real only) to ascertain the right applicable choice from provided formats.