Final answer:
The executive branch can check the legislative branch's power through veto power, influencing judicial appointments, and submitting amicus briefs. Congress can override vetoes with a two-thirds majority, creating a system of checks and balances.
Step-by-step explanation:
The executive branch can use its power to check the power of the legislative branch primarily through the veto power. If both chambers, the Senate and the House of Representatives, agree on a bill, it is then sent to the President for approval. The President has the authority to either sign the bill into law or veto it, providing a check on legislative power. However, Congress can override a presidential veto with a two-thirds supermajority, which serves as a check within a check.
Apart from the veto power, the executive branch, particularly the President, can also exert influence through the nomination of federal judges and through the submission of amicus briefs in legal cases, affecting the judiciary's interpretation and application of the law.
In a system of checks and balances, such powers help to ensure that no single branch of government, including the executive branch, can dominate the others, maintaining a balance of power within the U.S. government.