伸びる
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(VI) Stretch or physically extend – whether it’s a piece of silly putty, or a forest that extends all the way to the ocean, or days of summer that get longer. Use this one for hair! 髪が伸びたね! You grew your hair out!
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延びる
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(VI) To lengthen. Like 伸びる, but less often used. In fact, this 延びる is usually used only in one expression: 生き延びる (meaning, to survive a dangerous event, or lengthen one’s life)
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延ばす
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(VT) To lengthen or extend (the life of a sick person)(the deadline) (your stay in a hotel) also, to procrastinate. More used with time.
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伸ばす
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(VT) To lengthen – more used with physical things:stretch your back, extend a collapsible umbrella/telescope/ admantium claw, extend your talent by developing it.
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延長
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Two meanings: 1) an extension or prolonging, in time. 2) To extend a linear thing by building it longer. More official than just saying 延ばす.
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