Democrats played the “gender card” on Donald Trump with full force Tuesday, as President Obama described Hillary Clinton as a victim of sexism and Mrs. Clinton assailed the Republican nominee in a speech and on the airwaves over his alleged mistreatment of women.

With just one week until Election Day, Mr. Obama introduced a new theme into the race that has tightened amid a new FBI investigation of the Democratic nominee’s emails, telling voters in Ohio that Mrs. Clinton is being scrutinized unfairly in general because she’s a woman.

“Hillary Clinton is consistently treated differently than just about any other candidate I see out there,” Mr. Obama said at a rally at Capital University in Columbus.

Addressing male voters who favor Mr. Trump by significant margins in polls, Mr. Obama implored, “I just want to say to the guys out there There’s a reason we haven’t had a woman president before.”

“I think that sometimes we’re kind of trying to get over the hump,” Mr. Obama said. “I want every man out there who’s voting to kind of look inside yourself and ask yourself if you’re having problems with this stuff, how much of it is, you know, that we’re just not used to it.”

Mrs. Clinton, appearing at a campaign rally in Florida, was joined by former Miss Universe Alicia Machado, who became a major figure in the race after Mr. Trump targeted her on Twitter last month. Ms. Machado, now a vocal Clinton supporter, recalled Mr. Trump calling her an “eating machine” and “Miss Piggy” after she gained weight following the 1996 pageant.

“I was scared of him. He made fun of me and I didn’t know how to respond. He told me that I look ugly,” she told the crowd Tuesday. “It was really painful for me. He was cruel He thinks he can do whatever he wants and get away with it.”

In her own speech, Mrs. Clinton made no mention of the renewed FBI investigation into her use of a private email server. Instead, she focused on Mr. Trump and made a targeted appeal to women voters who, Mrs. Clinton argues, should be repulsed by what the Republican presidential nominee has said and done.

“If you’ve got a daughter, a granddaughter, a sister, a mother, a wife, a good friend, someone like this becoming president who insults more than half the population of the United States of America?” she said. “And what about our boys? This is not someone we want them looking up to.”

Polls consistently have shown Mrs. Clinton with a significant lead over Mr. Trump among female voters. Her Florida speech and introduction by Ms. Machado coincided with a new battleground-state TV ad that paints the Republican nominee as a misogynist.

Mr. Obama told the audience in Ohio that Mrs. Clinton is subjected to double standards as a female candidate.

“When a guy’s ambitious and out in the public arena, working hard, well that’s okay,” the president said. “But when a woman suddenly does it, you’re saying well why is she doing that? I’m just being honest. I want you to think about it because she is so much better qualified than the other guy.”

Seemingly referring to the email controversy, Mr. Obama said, “Has she made mistakes? Of course. So have I. There’s nobody in the public arena over the course of 30 years who doesn’t make some. But she is a fundamentally good and decent person who knows what she’s doing and will be an outstanding president.”

Earlier, the president had sounded a similar theme in an interview with late-night TV host Samantha Bee. He suggested that Mrs. Clinton’s email problems were a result of sexism.

Ms. Bee had asked the president what public perception Mrs. Clinton will face as president that would be the equivalent of the birther movement, which questioned Mr. Obama’s natural-born citizenship — and thus eligibility to be president — and was cited by liberals as simple racism.

“I think the equivalent will be: She’s tired, she’s moody, she’s being emotional, there’s something about her,” Mr. Obama said. “When men are ambitious, it’s just taken for granted. ‘Well, of course they should be ambitious.’ When women are ambitious, ‘Why?’ That theme, I think, will continue throughout her presidency, and it’s contributed to this notion that somehow she is hiding something.”

In the Clinton campaign’s new ad — which is airing in Arizona, Florida, Iowa, Nevada, North Carolina, Ohio and Pennsylvania — Mr. Trump’s own comments about women are played back over top of phrases like, “He really believes this.”

Mr. Trump’s now-infamous 2005 boast about being able to grope women because of his celebrity status is included in the commercial, as are other questionable statements.

“A person who is flat-chested is very hard to be a 10,” the businessman says in one clip highlighted in the ad.

The commercial also focuses heavily on the dozen woman who have accused Mr. Trump of sexual assault and closes with a message the Clinton campaign will be driving home throughout the final week of the campaign. “Anyone who believes, says, does what he does, is unfit to be president,” the ad says.

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