During the year ended December 31, 2022, we completed several strategic acquisitions to expand our solution offerings and enhance our customers' experience. The effects of the acquisitions were not material to our consolidated results of operations individually or in the aggregate. The combined purchase price of the acquisitions completed during the year ended December 31, 2022 was approximately $401.8 million, or $391.6 million net of cash acquired.

During the year ended December 31, 2022, we incurred acquisition-related expenses of $10.3 million. Acquisition-related expenses are recognized as selling, general and administrative and research and development expenses on the consolidated statements of income.

The assets acquired and liabilities assumed in connection with the acquisitions have been recorded based upon management's estimates of their fair market values as of each respective date of acquisition. The following tables summarize the fair value of consideration and the fair values of identified assets acquired and liabilities assumed for the combined 2022 acquisitions at each respective date of acquisition:


(in thousands)
Cash	$	10,242
Accounts receivable and other tangible assets	4,821
Developed software and core technologies (9 year weighted-average life)	127,830
Customer lists (14 year weighted-average life)	7,926
Trade names (10 year weighted-average life)	5,304
Accounts payable and other liabilities	(6,276)
Deferred revenue	(5,818)
Net deferred tax liabilities	(28,060)
Total identifiable net assets	$	115,969
Goodwill	$	285,877

The goodwill, which is not tax-deductible, is attributed to intangible assets that do not qualify for separate recognition, including the assembled workforces of the acquired businesses and the synergies expected to arise as a result of the acquisitions.

The fair values of the assets acquired and liabilities assumed are based on preliminary calculations. The estimates and assumptions for these items are subject to change as additional information about what was known and knowable at each respective acquisition date is obtained during the measurement period (up to one year from the acquisition date).

We determined the fair value of our intangible assets using various valuation techniques, including the relief-from-royalty method and the multi-period excess earnings method. These models utilize certain unobservable inputs classified as Level 3 measurements as defined by ASC 820, Fair Value Measurements and Disclosures. The determination of fair value requires considerable judgment and is sensitive to changes in underlying assumptions, estimates and market factors. Estimating fair value requires us to make assumptions and estimates regarding our future plans, as well as industry and economic conditions. These assumptions and estimates include, but are not limited to: selection of a valuation methodology, royalty rate, discount rate and attrition rate.

The weighted-average useful life, valuation method and assumptions used to determine the fair value of the intangible assets acquired in 2022 are as follows:

Intangible Asset

Weighted-Average Useful Life

Valuation Method

Assumptions

Developed software and core technologies

9 years

Multi-period excess earnings

Discount rate: 9.5% - 18.0%

Trade names

10 years

Relief-from-royalty

Royalty rate: 1.0% - 2.0%

Discount rate: 10.0% - 18.0%

Customer lists

14 years

Multi-period excess earnings

Attrition rate: 5.0% - 30.0%

Discount rate: 9.5% - 15.0%

The operating results of each acquisition have been included in our consolidated financial statements since each respective date of acquisition. The effects of the business combinations were not material to our consolidated results of operations individually or in the aggregate during 2022.

On October 1, 2021, we acquired 100% of the shares of Zemax, a leader in high-performance optical imaging system simulation, for a purchase price of $411.5 million, paid in cash, or $399.1 million net of cash acquired from Zemax. The acquisition expands the scope of our optical and photonics simulation portfolio by giving users comprehensive solutions that could drive innovation in healthcare, autonomy, consumer electronics and the IIoT.

Additionally, during the year ended December 31, 2021 we completed several other acquisitions to expand our solution offerings and enhance our customers' experience. These acquisitions were not individually significant. The combined purchase price of these acquisitions during the year ended December 31, 2021 was $110.7 million which was paid in cash, or $106.4 million net of cash acquired.

During the year ended December 31, 2021, we incurred $6.0 million in acquisition-related expenses, recognized as selling, general and administrative expense on the consolidated statements of income.

The assets and liabilities of the acquisitions have been recorded based upon management's estimates of their fair market values as of each respective date of acquisition. The following tables summarize the fair values of consideration transferred and the fair values of identified assets acquired and liabilities assumed at each respective date of acquisition:


(in thousands)	Zemax		Other Acquisitions		Total
Cash	$	12,353	$	4,320	$	16,673
Accounts receivable and other tangible assets	6,831		2,978		$	9,809
Developed software and core technologies (11 year weighted-average life)	96,000		32,200		$	128,200
Customer lists (8 year weighted-average life)	10,000		2,300		$	12,300
Trade names (10 year weighted-average life)	7,000		1,000		$	8,000
Accounts payable and other liabilities	(4,915)		(2,942)		$	(7,857)
Deferred revenue	(3,085)		(746)		$	(3,831)
Net deferred tax liabilities	(24,171)		(6,056)		$	(30,227)
Total identifiable net assets	$	100,013	$	33,054	$	133,067
Goodwill	$	311,488	$	77,685	$	389,173

The goodwill, which is generally not tax-deductible, is attributed to intangible assets that do not qualify for separate recognition, including the assembled workforce of the acquired businesses and the synergies expected to arise as a result of the acquisitions.

The weighted-average useful life, valuation method and assumptions used to determine the fair value of the intangible assets acquired with the Zemax acquisition are as follows:

Intangible Asset

Weighted-Average Useful Life

Valuation Method

Assumptions

Developed software and core technologies

11 years

Multi-period excess earnings

Discount rate: 7.5%

Trade names

10 years

Relief-from-royalty

Royalty rate: 2.0%

Discount rate: 8.0%

Customer lists

8 years

Multi-period excess earnings

Attrition rate: 10.0%

Discount rate: 7.5%