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an instrument with a bladed head on a handle or helve, used for hewing, cleaving, chopping, etc.
Jazz Slang. any musical instrument.
the ax, Informal.

dismissal from employment:
to get the ax.
expulsion from school.
rejection by a lover, friend, etc.:
His girlfriend gave him the ax.
any usually summary removal or curtailment.

to shape or trim with an ax.
to chop, split, destroy, break open, etc., with an ax:
The firemen had to ax the door to reach the fire.
Informal. to dismiss, restrict, or destroy brutally, as if with an ax:
The main office axed those in the field who didn’t meet their quota. Congress axed the budget.
Also, axe.
have an ax to grind, to have a personal or selfish motive:
His interest may be sincere, but I suspect he has an ax to grind.
ax.
Contemporary Examples

The axing has inspired countless articles on appropriate dismissal methods.
How to Get Fired in Spectacular Fashion Nina Strochlic August 21, 2013

Smith has said she got the impression that her axing came from the suits upstairs, not from show creator Shonda Rhimes.
What to Watch on TV This Week Nicole Ankowski November 8, 2008

Historical Examples

“She’d be axing the ladies to go home wid her and cook it for her after giving it her,” said another.
Castle Richmond Anthony Trollope

No, then, indeed I didn’t, darling; and glad enough he was to be axing me.
The Macdermots of Ballycloran Anthony Trollope

Like —— and blazes, your honour, axing your pardon, and the deck’s.
Hard Cash Charles Reade

There’s a jintleman in the drawing-room as is axing afther masther.
The Kellys and the O’Kellys Anthony Trollope

Had my eye pick’d out by a pavior, who was axing his way, he didn’t care where.
The Mirror of Literature, Amusement, and Instruction Various

You was axing, Tom, ef there was more business with thim Injun critters.
Indian and Scout F. S. Brereton

Give a man yer shirt, and ye must cut yer lucky or he’ll be after axing ye for yer skin.
The Bondman Hall Caine

Well, when this lad’s gone aboard, and is away, I’ll be axing for more of his gold.
With Wellington in Spain F. S. Brereton

noun (pl) axes
a hand tool with one side of its head forged and sharpened to a cutting edge, used for felling trees, splitting timber, etc See also hatchet
an axe to grind

an ulterior motive
a grievance
a pet subject

(informal) the axe

dismissal, esp from employment; the sack (esp in the phrase get the axe)
(Brit) severe cutting down of expenditure, esp the removal of unprofitable sections of a public service

(US, slang) any musical instrument, esp a guitar or horn
verb (transitive)
to chop or trim with an axe
(informal) to dismiss (employees), restrict (expenditure or services), or terminate (a project)
n.

Old English æces (Northumbrian acas) “axe, pickaxe, hatchet,” later æx, from Proto-Germanic *akusjo (cf. Old Saxon accus, Old Norse ex, Old Frisian axe, German Axt, Gothic aqizi), from PIE *agw(e)si- (cf. Greek axine, Latin ascia).

The spelling ax is better on every ground, of etymology, phonology, and analogy, than axe, which became prevalent during the 19th century; but it is now disused in Britain. [OED]

The spelling ax, though “better on every ground, of etymology, phonology, & analogy” (OED), is so strange to 20th-c. eyes that it suggests pedantry & is unlikely to be restored. [Fowler]

Meaning “musical instrument” is 1955, originally jazz slang for the saxophone; rock slang for “guitar” dates to 1967. The axe in figurative sense of cutting of anything (expenses, workers, etc.), especially as a cost-saving measure, is from 1922, probably from the notion of the headman’s literal axe (itself attested from mid-15c.). To have an axe to grind is from an 1815 essay by U.S. editor and politician Charles Miner (1780-1865) in which a man flatters a boy and gets him to do the chore of axe-grinding for him, then leaves without offering thanks or recompense. Misattributed to Benjamin Franklin in Weekley, OED print edition, and many other sources.

v.

1670s, “to shape or cut with an axe,” from axe (n.). Meaning “to remove, severely reduce,” usually figurative, recorded by 1922. Related: Axed; axing.
n.

see axe (n.).

ax abbr.
axis

noun

Any musical instrument, esp the saxophone: He played his ax at the casino (1950s+ Jazz musicians)
A guitar (Rock and roll)

verb

To dismiss someone from a job, a team, a school, a relationship, etc; can, fire: who suggested to Reagan that Deaver be axed
To eliminate; cut: They axed a lot of useless stuff from the budget

[musical instrument sense fr the resemblance in shape between a saxophone and an ax, and possibly fr the rhyme with sax]
ask (shortwave transmission)

used in the Authorized Version of Deut. 19:5; 20:19; 1 Kings 6:7, as the translation of a Hebrew word which means “chopping.” It was used for felling trees (Isa. 10:34) and hewing timber for building. It is the rendering of a different word in Judg. 9:48, 1 Sam. 13:20, 21, Ps. 74:5, which refers to its sharpness. In 2 Kings 6:5 it is the translation of a word used with reference to its being made of iron. In Isa. 44:12 the Revised Version renders by “axe” the Hebrew _maatsad_, which means a “hewing” instrument. In the Authorized Version it is rendered “tongs.” It is also used in Jer. 10:3, and rendered “axe.” The “battle-axe” (army of Medes and Persians) mentioned in Jer. 51:20 was probably, as noted in the margin of the Revised Version, a “maul” or heavy mace. In Ps. 74:6 the word so rendered means “feller.” (See the figurative expression in Matt. 3:10; Luke 3:9.)

In addition to the idiom beginning with ax