Learn Russian: При помощи призмы мы делим белый свет на спектр. - With the help of a prism, we divide white light into a spectrum.

They mean exactly the same thing ('with the help of' or 'using') and both require the Genitive case. However, при помощи sounds slightly more formal and is more common in technical, scientific, or official contexts, whereas с помощью is used in everyday speech as well as formal writing.

Could we just use the bare Instrumental case (призмой) instead of при помощи призмы?

Yes, you could say мы делим белый свет призмой. The core meaning remains the same. Using a formal compound preposition like при помощи or посредством simply makes the sentence sound more scientific, drawing extra attention to the method or instrument being used.

Why does призмы take the Genitive case?

The compound preposition при помощи is built around the noun помощь (help). When you link another noun to it (the help of a prism), that second noun must be in the Genitive case to show possession or association, which gives us the Genitive form призмы.

Why do we use на спектр (with на) instead of в спектр to mean 'into a spectrum'?

In Russian, when you divide, cut, or split something into parts, you use the verb делить (or similar verbs) with the preposition на plus the Accusative case. It is a set pattern for division, contrasting with the English use of the preposition 'into'.